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Martin Thoma
parent
efca80800b
commit
71f296ac0b
11 changed files with 884 additions and 6 deletions
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|||
#inner-details {
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||||
font-size:12px;
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||||
}
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||||
|
||||
span.close {
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||||
color:#FF5555;
|
||||
cursor:pointer;
|
||||
font-weight:bold;
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||||
margin-left:3px;
|
||||
}
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||||
|
||||
span.name {
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||||
cursor: pointer;
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||||
}
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||||
|
||||
/*TOOLTIPS*/
|
||||
.tip {
|
||||
text-align: left;
|
||||
width:auto;
|
||||
max-width:500px;
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||||
}
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||||
|
||||
.tip-title {
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||||
font-size: 11px;
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||||
text-align:center;
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||||
margin-bottom:2px;
|
||||
}
|
||||
123
presentations/Diskrete-Mathematik/Graph_files/base.css
Normal file
123
presentations/Diskrete-Mathematik/Graph_files/base.css
Normal file
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@ -0,0 +1,123 @@
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|||
html, body {
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||||
margin:0;
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||||
padding:0;
|
||||
font-family: "Lucida Grande", Verdana;
|
||||
font-size: 0.9em;
|
||||
text-align: center;
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||||
background-color:#F2F2F2;
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||||
}
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||||
|
||||
input, select {
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||||
font-size:0.9em;
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||||
}
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||||
|
||||
table {
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||||
margin-top:-10px;
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||||
margin-left:7px;
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||||
}
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||||
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||||
h4 {
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||||
font-size:1.1em;
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||||
text-decoration:none;
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||||
font-weight:normal;
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||||
color:#23A4FF;
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||||
}
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||||
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||||
a {
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||||
color:#23A4FF;
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||||
}
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||||
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||||
#container {
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||||
width: 1000px;
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||||
height: 600px;
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||||
margin:0 auto;
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||||
position:relative;
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||||
}
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||||
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||||
#left-container,
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#right-container,
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||||
#center-container {
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||||
height:600px;
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||||
position:absolute;
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||||
top:0;
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||||
}
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||||
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||||
#left-container, #right-container {
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||||
width:200px;
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||||
color:#686c70;
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||||
text-align: left;
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||||
overflow: auto;
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||||
background-color:#fff;
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||||
background-repeat:no-repeat;
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||||
border-bottom:1px solid #ddd;
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||||
}
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||||
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||||
#left-container {
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left:0;
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||||
background-image:url('col2.png');
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||||
background-position:center right;
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||||
border-left:1px solid #ddd;
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||||
|
||||
}
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||||
|
||||
#right-container {
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right:0;
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||||
background-image:url('col1.png');
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||||
background-position:center left;
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||||
border-right:1px solid #ddd;
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}
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||||
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||||
#right-container h4{
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||||
text-indent:8px;
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||||
}
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||||
|
||||
#center-container {
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||||
width:600px;
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||||
left:200px;
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||||
background-color:#1a1a1a;
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||||
color:#ccc;
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||||
}
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.text {
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margin: 7px;
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}
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#inner-details {
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font-size:0.8em;
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list-style:none;
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||||
margin:7px;
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||||
}
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||||
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#log {
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position:absolute;
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top:10px;
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font-size:1.0em;
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font-weight:bold;
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color:#23A4FF;
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}
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#infovis {
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position:relative;
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width:600px;
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height:600px;
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margin:auto;
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overflow:hidden;
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}
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||||
|
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/*TOOLTIPS*/
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.tip {
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color: #111;
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width: 139px;
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background-color: white;
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border:1px solid #ccc;
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-moz-box-shadow:#555 2px 2px 8px;
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-webkit-box-shadow:#555 2px 2px 8px;
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-o-box-shadow:#555 2px 2px 8px;
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box-shadow:#555 2px 2px 8px;
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||||
opacity:0.9;
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||||
filter:alpha(opacity=90);
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font-size:10px;
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font-family:Verdana, Geneva, Arial, Helvetica, sans-serif;
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padding:7px;
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}
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458
presentations/Diskrete-Mathematik/Graph_files/graph.js
Normal file
458
presentations/Diskrete-Mathematik/Graph_files/graph.js
Normal file
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var labelType, useGradients, nativeTextSupport, animate;
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(function() {
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var ua = navigator.userAgent,
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iStuff = ua.match(/iPhone/i) || ua.match(/iPad/i),
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typeOfCanvas = typeof HTMLCanvasElement,
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nativeCanvasSupport = (typeOfCanvas == 'object' || typeOfCanvas == 'function'),
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textSupport = nativeCanvasSupport
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&& (typeof document.createElement('canvas').getContext('2d').fillText == 'function');
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//I'm setting this based on the fact that ExCanvas provides text support for IE
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//and that as of today iPhone/iPad current text support is lame
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labelType = (!nativeCanvasSupport || (textSupport && !iStuff))? 'Native' : 'HTML';
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nativeTextSupport = labelType == 'Native';
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useGradients = nativeCanvasSupport;
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animate = !(iStuff || !nativeCanvasSupport);
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})();
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var Log = {
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elem: false,
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write: function(text){
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if (!this.elem)
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this.elem = document.getElementById('log');
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this.elem.innerHTML = text;
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this.elem.style.left = (500 - this.elem.offsetWidth / 2) + 'px';
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}
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};
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function init(){
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// init data
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var json = [
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{
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"adjacencies": [
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"graphnode0",
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{
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"nodeTo": "graphnode1",
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"nodeFrom": "graphnode0",
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"data": {
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"$color": "#557EAA"
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}
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}, {
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"nodeTo": "graphnode5",
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"nodeFrom": "graphnode0",
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"data": {
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"$color": "#909291"
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}
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}, {
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"nodeTo": "graphnode9",
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"nodeFrom": "graphnode0",
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"data": {
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"$color": "#557EAA"
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}
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}
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],
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"data": {
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"$color": "#83548B",
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"$type": "circle",
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"$dim": 8
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},
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"id": "graphnode0",
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"name": "graphnode0"
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}, {
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"adjacencies": [
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"graphnode1",
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{
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"nodeTo": "graphnode8",
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"nodeFrom": "graphnode1",
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"data": {
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"$color": "#557EAA"
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}
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}, {
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"nodeTo": "graphnode0",
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"nodeFrom": "graphnode1",
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"data": {
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"$color": "#909291"
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}
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}, {
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"nodeTo": "graphnode2",
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"nodeFrom": "graphnode1",
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"data": {
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"$color": "#557EAA"
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}
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||||
}
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||||
],
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"data": {
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||||
"$color": "#83548B",
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"$type": "circle",
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"$dim": 8
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},
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"id": "graphnode1",
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"name": "graphnode1"
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||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode2",
|
||||
{
|
||||
"nodeTo": "graphnode1",
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||||
"nodeFrom": "graphnode2",
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||||
"data": {
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||||
"$color": "#557EAA"
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||||
}
|
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}, {
|
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"nodeTo": "graphnode3",
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||||
"nodeFrom": "graphnode2",
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||||
"data": {
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||||
"$color": "#909291"
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||||
}
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||||
}, {
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||||
"nodeTo": "graphnode6",
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||||
"nodeFrom": "graphnode2",
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||||
"data": {
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||||
"$color": "#557EAA"
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||||
}
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||||
}
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||||
],
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"data": {
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||||
"$color": "#83548B",
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"$type": "circle",
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||||
"$dim": 8
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||||
},
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||||
"id": "graphnode2",
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||||
"name": "graphnode2"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode3",
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||||
{
|
||||
"nodeTo": "graphnode2",
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||||
"nodeFrom": "graphnode3",
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||||
"data": {
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||||
"$color": "#557EAA"
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||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode4",
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||||
"nodeFrom": "graphnode3",
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||||
"data": {
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||||
"$color": "#909291"
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||||
}
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||||
}, {
|
||||
"nodeTo": "graphnode9",
|
||||
"nodeFrom": "graphnode3",
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||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
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||||
"$type": "circle",
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||||
"$dim": 8
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||||
},
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||||
"id": "graphnode3",
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||||
"name": "graphnode3"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode4",
|
||||
{
|
||||
"nodeTo": "graphnode3",
|
||||
"nodeFrom": "graphnode4",
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||||
"data": {
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||||
"$color": "#557EAA"
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||||
}
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||||
}, {
|
||||
"nodeTo": "graphnode5",
|
||||
"nodeFrom": "graphnode4",
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||||
"data": {
|
||||
"$color": "#909291"
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||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode8",
|
||||
"nodeFrom": "graphnode4",
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||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
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||||
},
|
||||
"id": "graphnode4",
|
||||
"name": "graphnode4"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode5",
|
||||
{
|
||||
"nodeTo": "graphnode4",
|
||||
"nodeFrom": "graphnode5",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode0",
|
||||
"nodeFrom": "graphnode5",
|
||||
"data": {
|
||||
"$color": "#909291"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode6",
|
||||
"nodeFrom": "graphnode5",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
|
||||
},
|
||||
"id": "graphnode5",
|
||||
"name": "graphnode5"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode6",
|
||||
{
|
||||
"nodeTo": "graphnode5",
|
||||
"nodeFrom": "graphnode6",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode2",
|
||||
"nodeFrom": "graphnode6",
|
||||
"data": {
|
||||
"$color": "#909291"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode7",
|
||||
"nodeFrom": "graphnode6",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
|
||||
},
|
||||
"id": "graphnode6",
|
||||
"name": "graphnode6"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode7",
|
||||
{
|
||||
"nodeTo": "graphnode6",
|
||||
"nodeFrom": "graphnode7",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode8",
|
||||
"nodeFrom": "graphnode7",
|
||||
"data": {
|
||||
"$color": "#909291"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode9",
|
||||
"nodeFrom": "graphnode7",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
|
||||
},
|
||||
"id": "graphnode7",
|
||||
"name": "graphnode7"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode8",
|
||||
{
|
||||
"nodeTo": "graphnode1",
|
||||
"nodeFrom": "graphnode8",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode4",
|
||||
"nodeFrom": "graphnode8",
|
||||
"data": {
|
||||
"$color": "#909291"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode7",
|
||||
"nodeFrom": "graphnode8",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
|
||||
},
|
||||
"id": "graphnode8",
|
||||
"name": "graphnode8"
|
||||
}, {
|
||||
"adjacencies": [
|
||||
"graphnode9",
|
||||
{
|
||||
"nodeTo": "graphnode0",
|
||||
"nodeFrom": "graphnode9",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode3",
|
||||
"nodeFrom": "graphnode9",
|
||||
"data": {
|
||||
"$color": "#909291"
|
||||
}
|
||||
}, {
|
||||
"nodeTo": "graphnode7",
|
||||
"nodeFrom": "graphnode9",
|
||||
"data": {
|
||||
"$color": "#557EAA"
|
||||
}
|
||||
}
|
||||
],
|
||||
"data": {
|
||||
"$color": "#83548B",
|
||||
"$type": "circle",
|
||||
"$dim": 8
|
||||
},
|
||||
"id": "graphnode9",
|
||||
"name": "graphnode9"
|
||||
}
|
||||
];
|
||||
// end
|
||||
// init ForceDirected
|
||||
var fd = new $jit.ForceDirected({
|
||||
//id of the visualization container
|
||||
injectInto: 'infovis',
|
||||
//Enable zooming and panning
|
||||
//by scrolling and DnD
|
||||
Navigation: {
|
||||
enable: true,
|
||||
//Enable panning events only if we're dragging the empty
|
||||
//canvas (and not a node).
|
||||
panning: 'avoid nodes',
|
||||
zooming: 10 //zoom speed. higher is more sensible
|
||||
},
|
||||
// Change node and edge styles such as
|
||||
// color and width.
|
||||
// These properties are also set per node
|
||||
// with dollar prefixed data-properties in the
|
||||
// JSON structure.
|
||||
Node: {
|
||||
overridable: true
|
||||
},
|
||||
Edge: {
|
||||
overridable: true,
|
||||
color: '#23A4FF',
|
||||
lineWidth: 0.4
|
||||
},
|
||||
//Native canvas text styling
|
||||
Label: {
|
||||
type: labelType, //Native or HTML
|
||||
size: 10,
|
||||
style: 'bold'
|
||||
},
|
||||
//Add Tips
|
||||
Tips: {
|
||||
enable: true,
|
||||
onShow: function(tip, node) {
|
||||
//count connections
|
||||
var count = 0;
|
||||
node.eachAdjacency(function() { count++; });
|
||||
//display node info in tooltip
|
||||
tip.innerHTML = "<div class=\"tip-title\">" + node.name + "</div>"
|
||||
+ "<div class=\"tip-text\"><b>connections:</b> " + count + "</div>";
|
||||
}
|
||||
},
|
||||
// Add node events
|
||||
Events: {
|
||||
enable: true,
|
||||
type: 'Native',
|
||||
//Change cursor style when hovering a node
|
||||
onMouseEnter: function() {
|
||||
fd.canvas.getElement().style.cursor = 'move';
|
||||
},
|
||||
onMouseLeave: function() {
|
||||
fd.canvas.getElement().style.cursor = '';
|
||||
},
|
||||
//Update node positions when dragged
|
||||
onDragMove: function(node, eventInfo, e) {
|
||||
var pos = eventInfo.getPos();
|
||||
node.pos.setc(pos.x, pos.y);
|
||||
fd.plot();
|
||||
},
|
||||
//Implement the same handler for touchscreens
|
||||
onTouchMove: function(node, eventInfo, e) {
|
||||
$jit.util.event.stop(e); //stop default touchmove event
|
||||
this.onDragMove(node, eventInfo, e);
|
||||
},
|
||||
//Add also a click handler to nodes
|
||||
onClick: function(node) {
|
||||
if(!node) return;
|
||||
// Build the right column relations list.
|
||||
// This is done by traversing the clicked node connections.
|
||||
var html = "<h4>" + node.name + "</h4><b> connections:</b><ul><li>",
|
||||
list = [];
|
||||
node.eachAdjacency(function(adj){
|
||||
list.push(adj.nodeTo.name);
|
||||
});
|
||||
//append connections information
|
||||
$jit.id('inner-details').innerHTML = html + list.join("</li><li>") + "</li></ul>";
|
||||
}
|
||||
},
|
||||
//Number of iterations for the FD algorithm
|
||||
iterations: 200,
|
||||
//Edge length
|
||||
levelDistance: 130,
|
||||
// Add text to the labels. This method is only triggered
|
||||
// on label creation and only for DOM labels (not native canvas ones).
|
||||
onCreateLabel: function(domElement, node){
|
||||
domElement.innerHTML = node.name;
|
||||
var style = domElement.style;
|
||||
style.fontSize = "0.8em";
|
||||
style.color = "#ddd";
|
||||
},
|
||||
// Change node styles when DOM labels are placed
|
||||
// or moved.
|
||||
onPlaceLabel: function(domElement, node){
|
||||
var style = domElement.style;
|
||||
var left = parseInt(style.left);
|
||||
var top = parseInt(style.top);
|
||||
var w = domElement.offsetWidth;
|
||||
style.left = (left - w / 2) + 'px';
|
||||
style.top = (top + 10) + 'px';
|
||||
style.display = '';
|
||||
}
|
||||
});
|
||||
// load JSON data.
|
||||
fd.loadJSON(json);
|
||||
// compute positions incrementally and animate.
|
||||
fd.computeIncremental({
|
||||
iter: 40,
|
||||
property: 'end',
|
||||
onStep: function(perc){
|
||||
Log.write(perc + '% loaded...');
|
||||
},
|
||||
onComplete: function(){
|
||||
Log.write('done');
|
||||
fd.animate({
|
||||
modes: ['linear'],
|
||||
transition: $jit.Trans.Elastic.easeOut,
|
||||
duration: 2500
|
||||
});
|
||||
}
|
||||
});
|
||||
// end
|
||||
}
|
||||
23
presentations/Diskrete-Mathematik/Graph_files/jit-yc.js
Normal file
23
presentations/Diskrete-Mathematik/Graph_files/jit-yc.js
Normal file
File diff suppressed because one or more lines are too long
Binary file not shown.
|
|
@ -28,6 +28,12 @@ Kantenmenge bezeichnet.
|
|||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Isomorphe Graphen}
|
||||
\begin{center}
|
||||
\href{http://www.martin-thoma.de/uni/graph.html}{martin-thoma.de/uni/graph.html}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Inzidenz}
|
||||
\begin{block}{Inzidenz}
|
||||
Sei $e \in E$ und $k = \Set{v_1, v_2} \in K$.
|
||||
|
|
|
|||
|
|
@ -26,21 +26,64 @@ Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält.
|
|||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\pgfdeclarelayer{background}
|
||||
\pgfsetlayers{background,main}
|
||||
\begin{frame}{Eulerscher Kreis}
|
||||
TODO: $K_5$ eulerkreis animieren
|
||||
\newcommand\n{5}
|
||||
\tikzstyle{selected edge} = [draw,line width=5pt,-,red!50]
|
||||
\begin{center}
|
||||
\adjustbox{max size={\textwidth}{0.8\textheight}}{
|
||||
\begin{tikzpicture}
|
||||
\foreach \number in {1,...,\n}{
|
||||
\node[vertex] (N-\number) at ({\number*(360/\n)}:5.4cm) {};
|
||||
}
|
||||
|
||||
\foreach \number in {1,...,\n}{
|
||||
\foreach \y in {1,...,\n}{
|
||||
\draw (N-\number) -- (N-\y);
|
||||
}
|
||||
}
|
||||
|
||||
\node<2->[vertex,red] (N-1) at ({1*(360/\n)}:5.4cm) {};
|
||||
|
||||
\begin{pgfonlayer}{background}
|
||||
\path<2->[selected edge] (N-1.center) edge node {} (N-2.center);
|
||||
\path<3->[selected edge] (N-2.center) edge node {} (N-3.center);
|
||||
\path<4->[selected edge] (N-3.center) edge node {} (N-4.center);
|
||||
\path<5->[selected edge] (N-4.center) edge node {} (N-5.center);
|
||||
\path<6->[selected edge] (N-5.center) edge node {} (N-1.center);
|
||||
\path<7->[selected edge] (N-1.center) edge node {} (N-3.center);
|
||||
\path<8->[selected edge] (N-3.center) edge node {} (N-5.center);
|
||||
\path<9->[selected edge] (N-5.center) edge node {} (N-2.center);
|
||||
\path<10->[selected edge] (N-2.center) edge node {} (N-4.center);
|
||||
\path<11->[selected edge](N-4.center) edge node {} (N-1.center);
|
||||
\end{pgfonlayer}
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\subsection{Satz von Euler}
|
||||
\begin{frame}{Satz von Euler}
|
||||
\begin{block}{Satz von Euler}
|
||||
Wenn ein Graph $G$ eulersch ist, dann hat jeder Knoten von $G$ geraden Grad.
|
||||
Wenn ein Graph $G$ eulersch ist, dann hat jede Ecke von $G$ geraden Grad.
|
||||
\end{block}
|
||||
|
||||
Wenn $G$ einen Knoten mit ungeraden Grad hat, ist $G$ nicht eulersch.
|
||||
\pause
|
||||
|
||||
$\Rightarrow$ Wenn $G$ eine Ecke mit ungeraden Grad hat, ist $G$ nicht eulersch.
|
||||
|
||||
\pause
|
||||
|
||||
\begin{gallery}
|
||||
\galleryimage{vollstaendig/k-5}
|
||||
\galleryimage{koenigsberg/koenigsberg-1}
|
||||
\end{gallery}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Umkehrung des Satzes von Euler}
|
||||
\begin{block}{Umkehrung des Satzes von Euler}
|
||||
Wenn in einem zusammenhängenden Graphen $G$ jeder Knoten geraden Grad hat, dann
|
||||
Wenn in einem zusammenhängenden Graphen $G$ jede Ecke geraden Grad hat, dann
|
||||
ist $G$ eulersch.
|
||||
\end{block}
|
||||
|
||||
|
|
@ -68,6 +111,54 @@ $G$ hat eine offene eulersche Linie $:\Leftrightarrow G$ hat genau zwei Ecken
|
|||
ungeraden Grades.
|
||||
\end{block}
|
||||
|
||||
TODO: Haus des Nikolaus-Animation.
|
||||
TODO: Beweis
|
||||
\pause
|
||||
|
||||
\begin{block}{Beweis "`$\Rightarrow"'$}
|
||||
Sei $G=(E, K)$ ein zusammenhängender Graph und $L = (e_0, \dots, e_s)$ eine offene
|
||||
eulersche Linie. \pause
|
||||
Sei $G^* = (E, K \cup \Set{e_s, e_0})$. \pause
|
||||
Es gibt einen Eulerkreis in $G^*$ \pause \\
|
||||
$\xRightarrow{\text{Satz von Euler}}$ In $G^*$ hat jede Ecke geraden Grad \pause \\
|
||||
Der Grad von nur zwei Kanten wurde um jeweils 1 erhöht \pause \\
|
||||
$\Rightarrow$ in $G$ haben genau 2 Ecken ungeraden Grad $\blacksquare$
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\pgfdeclarelayer{background}
|
||||
\pgfsetlayers{background,main}
|
||||
\begin{frame}{Haus des Nikolaus}
|
||||
\tikzstyle{selected edge} = [draw,line width=5pt,-,red!50]
|
||||
\begin{center}
|
||||
\adjustbox{max size={\textwidth}{0.8\textheight}}{
|
||||
\begin{tikzpicture}
|
||||
\node[vertex] (a) at (0,0) {};
|
||||
\node[vertex] (b) at (2,0) {};
|
||||
\node[vertex] (c) at (2,2) {};
|
||||
\node[vertex] (d) at (0,2) {};
|
||||
\node[vertex] (e) at (1,4) {};
|
||||
|
||||
\draw (a) -- (d);
|
||||
\draw (d) -- (b);
|
||||
\draw (b) -- (c);
|
||||
\draw (c) -- (d);
|
||||
\draw (d) -- (e);
|
||||
\draw (e) -- (c);
|
||||
\draw (c) -- (a);
|
||||
\draw (a) -- (b);
|
||||
|
||||
\node<2->[vertex, red] (a) at (0,0) {};
|
||||
|
||||
\begin{pgfonlayer}{background}
|
||||
\path<2->[selected edge] (a.center) edge node {} (d.center);
|
||||
\path<3->[selected edge] (d.center) edge node {} (b.center);
|
||||
\path<4->[selected edge] (b.center) edge node {} (c.center);
|
||||
\path<5->[selected edge] (c.center) edge node {} (d.center);
|
||||
\path<6->[selected edge] (d.center) edge node {} (e.center);
|
||||
\path<7->[selected edge] (e.center) edge node {} (c.center);
|
||||
\path<8->[selected edge] (c.center) edge node {} (a.center);
|
||||
\path<9->[selected edge] (a.center) edge node {} (b.center);
|
||||
\end{pgfonlayer}
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,21 @@
|
|||
\documentclass[varwidth=true, border=2pt]{standalone}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows,positioning}
|
||||
\tikzset{
|
||||
%Define standard arrow tip
|
||||
>=stealth',
|
||||
% Define arrow style
|
||||
pil/.style={->,thick}
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
\begin{tikzpicture}
|
||||
\node (a)[vertex] at (0,8) {$a$};
|
||||
\node (b)[vertex] at (0,4) {$b$};
|
||||
\node (c)[vertex] at (0,0) {$c$};
|
||||
\node (d)[vertex] at (4,4) {$d$};
|
||||
|
||||
\foreach \from/\to/\pos in {a/b/20,a/b/-20,a/d/0,b/c/20,b/c/-20,b/d/0,c/d/0}
|
||||
\draw[line width=2pt] (\from) to [bend left=\pos] (\to);
|
||||
\end{tikzpicture}
|
||||
\end{document}
|
||||
7
presentations/Diskrete-Mathematik/Plan/Makefile
Normal file
7
presentations/Diskrete-Mathematik/Plan/Makefile
Normal file
|
|
@ -0,0 +1,7 @@
|
|||
SOURCE = Plan
|
||||
make:
|
||||
pdflatex $(SOURCE).tex -output-format=pdf
|
||||
make clean
|
||||
|
||||
clean:
|
||||
rm -rf $(TARGET) *.class *.html *.log *.aux *.out
|
||||
121
presentations/Diskrete-Mathematik/Plan/Plan.tex
Normal file
121
presentations/Diskrete-Mathematik/Plan/Plan.tex
Normal file
|
|
@ -0,0 +1,121 @@
|
|||
\documentclass[a4paper,9pt]{scrartcl}
|
||||
\usepackage{amssymb, amsmath} % needed for math
|
||||
\usepackage[utf8]{inputenc} % this is needed for umlauts
|
||||
\usepackage[ngerman]{babel} % this is needed for umlauts
|
||||
\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
|
||||
\usepackage[margin=2.5cm]{geometry} %layout
|
||||
\usepackage{hyperref} % links im text
|
||||
\usepackage{color}
|
||||
\usepackage{framed}
|
||||
\usepackage{enumerate} % for advanced numbering of lists
|
||||
\clubpenalty = 10000 % Schusterjungen verhindern
|
||||
\widowpenalty = 10000 % Hurenkinder verhindern
|
||||
|
||||
\hypersetup{
|
||||
pdfauthor = {Martin Thoma},
|
||||
pdfkeywords = {Lineare Algebra},
|
||||
pdftitle = {Vortrag Graphentheorie I: Tafelbild + Text}
|
||||
}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Custom definition style, by %
|
||||
% http://mathoverflow.net/questions/46583/what-is-a-satisfactory-way-to-format-definitions-in-latex/58164#58164
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\makeatletter
|
||||
\newdimen\errorsize \errorsize=0.2pt
|
||||
% Frame with a label at top
|
||||
\newcommand\LabFrame[2]{%
|
||||
\fboxrule=\FrameRule
|
||||
\fboxsep=-\errorsize
|
||||
\textcolor{FrameColor}{%
|
||||
\fbox{%
|
||||
\vbox{\nobreak
|
||||
\advance\FrameSep\errorsize
|
||||
\begingroup
|
||||
\advance\baselineskip\FrameSep
|
||||
\hrule height \baselineskip
|
||||
\nobreak
|
||||
\vskip-\baselineskip
|
||||
\endgroup
|
||||
\vskip 0.5\FrameSep
|
||||
\hbox{\hskip\FrameSep \strut
|
||||
\textcolor{TitleColor}{\textbf{#1}}}%
|
||||
\nobreak \nointerlineskip
|
||||
\vskip 1.3\FrameSep
|
||||
\hbox{\hskip\FrameSep
|
||||
{\normalcolor#2}%
|
||||
\hskip\FrameSep}%
|
||||
\vskip\FrameSep
|
||||
}}%
|
||||
}}
|
||||
\definecolor{FrameColor}{rgb}{0.25,0.25,1.0}
|
||||
\definecolor{TitleColor}{rgb}{1.0,1.0,1.0}
|
||||
|
||||
\newenvironment{contlabelframe}[2][\Frame@Lab\ (cont.)]{%
|
||||
% Optional continuation label defaults to the first label plus
|
||||
\def\Frame@Lab{#2}%
|
||||
\def\FrameCommand{\LabFrame{#2}}%
|
||||
\def\FirstFrameCommand{\LabFrame{#2}}%
|
||||
\def\MidFrameCommand{\LabFrame{#1}}%
|
||||
\def\LastFrameCommand{\LabFrame{#1}}%
|
||||
\MakeFramed{\advance\hsize-\width \FrameRestore}
|
||||
}{\endMakeFramed}
|
||||
\newcounter{definition}
|
||||
\newenvironment{definition}[1]{%
|
||||
\par
|
||||
\refstepcounter{definition}%
|
||||
\begin{contlabelframe}{Definition \thedefinition:\quad #1}
|
||||
\noindent\ignorespaces}
|
||||
{\end{contlabelframe}}
|
||||
\makeatother
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Begin document %
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
\section{Königsberger Brückenproblem}
|
||||
\subsection{Beweis des Satzes von Euler}
|
||||
Tafelbild:
|
||||
|
||||
Sie $G = (E, K)$ ein eulerscher Graph, $K$ ein Eulerkreis durch $G$ und
|
||||
$e \in E$ eine beliebige Kante.
|
||||
Dann geht $K$ durch $e$. Nun sei $a$ die Anzahl, wie häufig $K$ durch $e$ geht.
|
||||
Offensichtlich geht der Kreis sowohl in den Knoten hinein, als auch hinaus.
|
||||
$\Rightarrow e$ hat mindestens den Knotengrad $2a$. Es kann keine weitere
|
||||
Kante geben, da jeder Eulerkreis zu $G$ alle Kanten von $G$ beinhaltet.
|
||||
$\Rightarrow e$ hat den Knotengrad $2a \Rightarrow$ Jede Ecke von $G$ hat geraden
|
||||
Grad. $\blacksquare$
|
||||
|
||||
\subsection{Rückrichtung}
|
||||
Hat jede Ecke in einem zusammenhängendem Graphen $G$ geraden Grad, so ist $G$ eulerisch.
|
||||
|
||||
Beweis durch Induktion über die Anzahl $m$ der Kanten.
|
||||
|
||||
\textbf{I.A.}:
|
||||
\begin{itemize}
|
||||
\item $m=0 \rightarrow$ trivial
|
||||
\item $m = 1$: nicht möglich
|
||||
\item $m = 2$: Da $G$ zusammenhängend ist, können in diesem Fall nur zwei
|
||||
Ecken zweifach miteinander verbunden sein $\Rightarrow$ auch eulersch
|
||||
\end{itemize}
|
||||
|
||||
\textbf{I.V.}: Sei $m \in \mathbb{N}_{\geq 2}$ die Anzahl der Kanten eines
|
||||
Graphs $G$ und jeder zusammenhängende Graph mit weniger als $m$ Kanten und
|
||||
ausschließlich Knoten geraden Grades sei eulerisch.
|
||||
|
||||
\textbf{I.S.}
|
||||
|
||||
Jeder Knoten hat mindestens Grad 2 (zusammenhängend + gerader Grad)
|
||||
$\Rightarrow$ es gibt einen Kreis in $G$. TODO
|
||||
|
||||
Sei nun $C$ ein Kreis in $G$ mit maximaler Länge.
|
||||
|
||||
Annahme: $C$ ist kein Eulerkreis
|
||||
|
||||
Wir entfernen alle Kanten in $C$ aus $G$ und nennen das Ergebnis $G^*$.
|
||||
Dann hat jeder Zusammenhängende Teilgraph in $G^*$ nur Knoten geraden Grades
|
||||
und hat daher einen Eulerkreis. Dieser Eulerkreis hat keine Kante, die in $C$
|
||||
enthalten ist und könnte deshalb zu $C$ hinzugefügt werden, wodurch $C$ Länger
|
||||
werden würde $\Rightarrow$ Widerspruch $\Rightarrow C$ ist ein Eulerkreis
|
||||
$\Rightarrow G$ ist eulersch $\blacksquare$
|
||||
|
||||
\end{document}
|
||||
|
|
@ -133,3 +133,4 @@
|
|||
|
||||
\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt]
|
||||
\def\TCop{\textsuperscript{\textcopyright}} % Copyright-sign
|
||||
\usepackage{mathtools}
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue