在图论中，Vizing定理指出，每个简单的无向图都可以使用最多比图的最大度数“ d”大一个的多种颜色进行边色着色。简单来说，该定理指出，简单图的色指数可以为d或d +1 。图形的边缘着色所需的最大颜色数称为色度指数。
上图中的G中有5个顶点。最高度为4，但我们需要5种颜色，因此没有边与相邻顶点的任何边共享相同的颜色，如上图所示。因此，此图所需的有效颜色数等于5，即(“最高度” +1)。
注意：上图中的c1，c2，c3，c4和c5表示不同的颜色。
例子 ：
Input :
v = 3, e = 3
{{ 1, 2, -1 },
{ 2, 3, -1 },
{ 3, 1, -1 }};
Output :
3 colors needs to generate a valid edge coloring :
color between v(1): 1 and v(2): 2 is: color 1
color between v(1): 2 and v(2): 3 is: color 2
color between v(1): 3 and v(2): 1 is: color 3
算法：
- 初始化边数后，分配每个边的顶点对
- 根据定理为图形边缘着色
- 分配颜色，然后检查其有效性
- 检查是否有两个相邻的边具有相同的颜色，然后增加颜色，转到标记并尝试下一个颜色
- 重复上述步骤，直到所有边缘都达到定理的颜色为止
- 完成后，打印顶点之间所有边缘的颜色
下面是上述方法的实现：
C++
// C++ program to illustrate
// Vizing's Theorem
#include
using namespace std;
// Function to print the color of the edge
void colorEdge(int edges[][3], int e)
{
int color;
// Assign a color to every edge 'i'.
for (int i = 0; i < e; i++) {
color = 1;
flag:
// Assign a color and
// then check its validity.
edges[i][2] = color;
for (int j = 0; j < e; j++) {
if (j == i)
continue;
// If the color of edges
// is adjacent to edge i
if ((edges[i][0] == edges[j][0])
|| (edges[i][1] == edges[j][0])
|| (edges[i][0] == edges[j][1])
|| (edges[i][1] == edges[j][1])) {
// If the color matches
if (edges[i][2] == edges[j][2]) {
// Increment the color,
// denotes the change in color.
color++;
// goes back, and assigns
// the next color.
goto flag;
}
}
}
}
// Check the maximum color from all the edge colors
int maxColor = -1;
for (int i = 0; i < e; i++) {
maxColor = max(maxColor, edges[i][2]);
}
cout << maxColor
<< " colors needs to generate a valid edge "
"coloring:"
<< endl;
for (int i = 0; i < e; i++) {
cout << "color between v(1): " << edges[i][0]
<< " and v(2): " << edges[i][1]
<< " is: color " << edges[i][2] << endl;
}
}
// Driver Code
int main()
{
// initialize the number
// of edges of the graph
int e = 5;
// initialize the vertex
// pair of every edge in a graph
int edges[e][3] = { { 1, 2, -1 },
{ 2, 3, -1 },
{ 3, 4, -1 },
{ 4, 1, -1 },
{ 1, 3, -1 } };
colorEdge(edges, e);
return 0;
} Java
// Java program to illustrate
// Vizing's Theorem
import java.util.*;
public class VizingsTheorem {
// Function to find the chromatic index
public void colorEdge(int[][] edges, int e)
{
// Initialize edge to first edge and
// color to color 1
int i = 0, color = 1;
// Repeat until all edges are done coloring
while (i < e) {
// Give the selected edge a color
edges[i][2] = color;
boolean flag = false;
// Iterate through all others edges to check
for (int j = 0; j < e; j++) {
// Ignore if same edge
if (j == i)
continue;
// Check if one vertex is similar
if ((edges[i][0] == edges[j][0])
|| (edges[i][1] == edges[j][0])
|| (edges[i][0] == edges[j][1])
|| (edges[i][1] == edges[j][1])) {
// Check if color is similar
if (edges[i][2] == edges[j][2]) {
// Increment the color by 1
color++;
flag = true;
break;
}
}
}
// If same color faced then repeat again
if (flag == true) {
continue;
}
// Or else proceed to a
// new vertex with color 1
color = 1;
i++;
}
// Check the maximum color from all the edge colors
int maxColor = -1;
for (i = 0; i < e; i++)
{
maxColor = Math.max(maxColor, edges[i][2]);
}
System.out.println(
maxColor
+ " colors needs to generate"
+" a valid edge coloring:");
for (i = 0; i < e; i++)
{
System.out.println(
"color between v(1): " + edges[i][0]
+ " and v(2): " + edges[i][1]
+ " is: color " + edges[i][2]);
}
}
// Driver code
public static void main(String[] args)
{
// Number of edges
int e = 5;
// Edge list
int[][] edges = new int[e][3];
// Initialize all edge colors to 0
for (int i = 0; i < e; i++) {
edges[i][2] = -1;
}
// Edges
edges[0][0] = 1;
edges[0][1] = 2;
edges[1][0] = 2;
edges[1][1] = 3;
edges[2][0] = 3;
edges[2][1] = 4;
edges[3][0] = 4;
edges[3][1] = 1;
edges[4][0] = 1;
edges[4][1] = 3;
// Run the function
VizingsTheorem c = new VizingsTheorem();
c.colorEdge(edges, e);
}
}Python3
def colorEdge(edges, e):
# Initialize edge to first edge and
# color to color 1
i = 0
color = 1
# Repeat until all edges are done coloring
while(i < e):
# Give the selected edge a color
edges[i][2] = color
flag = False
# Iterate through all others edges to check
for j in range(e):
# Ignore if same edge
if (j == i):
continue
# Check if one vertex is similar
if ((edges[i][0] == edges[j][0])or
(edges[i][1] == edges[j][0]) or
(edges[i][0] == edges[j][1]) or
(edges[i][1] == edges[j][1])):
# Check if color is similar
if (edges[i][2] == edges[j][2]):
# Increment the color by 1
color += 1
flag = True
break
# If same color faced then repeat again
if (flag == True):
continue
# Or else proceed to a new vertex with color 1
color = 1
i += 1
# Check the maximum color from all the edge colors
maxColor = -1
for i in range(e):
maxColor = max(maxColor, edges[i][2])
print(str(maxColor)+" colors needs to generate a valid edge coloring:")
for i in range(e):
print("color between v(1): "+str(edges[i][0])+" and v(2): "
+ str(edges[i][1])+" is: color "+str(edges[i][2]))
# Driver code
if __name__ == "__main__":
# Number of edges
e = 5
# Edge list
edges = [[0 for _ in range(3)] for _ in range(e)]
# Initialize all edge colors to 0
for i in range(e):
edges[i][2] = -1
# Edges
edges[0][0] = 1
edges[0][1] = 2
edges[1][0] = 2
edges[1][1] = 3
edges[2][0] = 3
edges[2][1] = 4
edges[3][0] = 4
edges[3][1] = 1
edges[4][0] = 1
edges[4][1] = 3
# Run the function
colorEdge(edges, e)输出
3 colors needs to generate a valid edge coloring:
color between v(1): 1 and v(2): 2 is: color 1
color between v(1): 2 and v(2): 3 is: color 2
color between v(1): 3 and v(2): 4 is: color 1
color between v(1): 4 and v(2): 1 is: color 2
color between v(1): 1 and v(2): 3 is: color 3