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In continuation of the previous question, what is a strict upper bound on the chromatic number of the square of a bipartite graph?

I think the chromatic number number of the square of the bipartite graph with maximum degree $\Delta=2$ and a cycle is at most $4$ and with $\Delta\ge3$ is at most $\Delta+1$. This is because the edge set of a connected bipartite graph consists of the edges of a union of trees and a edge disjoint union of even cycles (with or without chords). Now, the square of a cycle requires at most $4$ colors, and the square of a tree requires at most $\Delta+1$ colors. Thus, the required number of colors is $\Delta+1$ in the latter case. Am I right here? Any counterexamples? Thanks beforehand.

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The maximum degree of $G^2$ for general $G$ is at most $\Delta^2$, so we immetiately get an upper bound $\chi(G^2)\le \Delta^2+1$.

An example that is close to optimal is the incidence graph of the points and lines of a finite projective plane of order $q$. Here we have $2(q^2+q+1)$ vertices and the graph is regular of degree $\Delta=q+1$. The square of this graph has maximal cliques of size $q^2+q+1$ and in fact this is also equal to the chromatic number, so $\chi (G^2)=\Delta^2-\Delta+1$. In particular you shouldn't expect a linear bound in $\Delta$ without further conditions.

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  • $\begingroup$ thanks! well, I thought that the maximum degree of the square of a graph is twice that of the original graph, by looking at cases like the square of cycles and trees. What about the half-square graphs of a bipartite graph? I think at least in that case we can bound the chromatic number in linear way $\endgroup$ – vidyarthi Jul 3 at 5:34
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    $\begingroup$ The finite projective plane gives a quadratic chromatic number even if you just take the half square. $\endgroup$ – Gjergji Zaimi Jul 3 at 5:37
  • $\begingroup$ Would it be of any use if the major degree vertices induce a forest, like in $K_{m,n}\ \ m\neq n$? $\endgroup$ – vidyarthi Jul 3 at 11:48

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