Prove that a graph is bipartite

Question

Given a graph G in which every edge connects an even degree node with an odd degree node. How can i prove that the graph is bipartite?

Thanks in advance

Answer 1

Because by definition you already have two disjoint sets of vertices such that the only edges go between a vertex in one set and a vertex in the other set.

The even degree nodes are one set, and the odd degree nodes are the other set.

Answer 2

Pick any node, put it in set A. Take all the nodes that link to it, put them in set B. Now for every node added, add all it's neighbors to the opposite set, and check for the ones that already belong to one of the sets that they are in the right set. If you get a contradiction then the graph is not bipartite.

If you run out of neighbors but there are still nodes left, pick again any node and continue the algorithm until you either have no nodes or you found a contradiction.

Answer 3

This is the Welsh-Powell graph colouring algorithm:

1. All vertices are sorted according to the decreasing value of their degree in a list V

2. Colours are ordered in a list C

3. The first non-coloured vertex v in V is coloured with the first available colour in C. "Available" means a colour that has not previously used by this algorithm

4. The remaining part of the ordered list V is traversed and the same colour is allocated to every vertex for which no adjacent vertex has the same colour

5. Steps 3 and 4 are applied iteratively until all the vertices have been coloured

A graph is bipartite if it is 2-colourable. This intuitive fact is proven in Cambridge Tracts in Mathematics 131.

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This is, of course, the cannon with which to shoot a mosquito. A graph is bipartite iff its vertices can be divided into two sets, such that every edge connects a vertex from set 1 to one in set 2. You already have such a division: each edge connects a vertex from the set of odd-degree vertices, to a vertex in the set of even-degree vertices.

Answer 4

If by "prove", you mean "find out", the complete solution is here

Bipartite.java

It works by keeping two boolean arrays. A marked array to check if a node has been visited, and another colored array. It then does a depth first search, marking neighbors with alternate colors. If a neighbor is marked with same color, graph is not bipartite.