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The title says it all. Let $G$ be a (simple undirected finite) graph, and let $G^c$ be the complement of $G$ (i.e. contains edges not in $G$ and no more). Let $M$ be a perfect matching in $G^c$.
How large can the difference between the chromatic number of $G$ and the chromatic number of $G+M$ be?
That is, can we say anything about $\max_M \chi(G+M)-\chi(G)$? (at least some bounds).

I would like to keep the problem in the general setting. If it matters, I am more interested in the special case when $G$ is a regular graph.

Thank you.

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    $\begingroup$ What kind of bounds do you have in mind? In general, I don’t think any bound on the difference can beat the original chromatic number itself, i.e I think for every $k$, you can find a regular graph with chromatic number $k$ such that adding a perfect matching causes it to double (consider complete $k$-partite graphs and add the edges within each part). This should be tight—given the original optimal coloring, make two new colors for each original color to get a valid coloring to fix any edge from the perfect matching connecting two vertices of same original color. But I could be wrong... $\endgroup$
    – Jason Gaitonde
    Nov 15, 2020 at 6:12
  • $\begingroup$ @J.G Okay. I think your answer is correct. Please post it as an answer. $\endgroup$
    – Cyriac Antony
    Nov 15, 2020 at 6:40

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One always has $\chi(G+M)\leq 2\chi(G)$, and this bound does not seem improvable in general. To see the bound itself, note that the only edges of the matching $M$ that can possibly ruin an optimal coloring of $G$ are those between two vertices of the same color class; in particular, we can color $G+M$ by simply creating two new colors for every original color and placing the two nodes on any such violating edge with distinct colors.

To see that this is tight, for any $k$, consider a $k$-partite complete graph with an even number of nodes in each part. It is easy to see that the original graph is $k$-colorable and that adding any perfect matching (which only can have edges within each part) will increase the clique number to $2k$, and so the chromatic number is at least $2k$.

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  • $\begingroup$ Are there any simple structures that ensure that the difference $\chi(G+M)-\chi(G)$ is small? eg: $G$ is $2K_2$-free. $\endgroup$
    – Cyriac Antony
    Nov 17, 2020 at 3:27
  • $\begingroup$ @CyriacAntony no idea, but that's a nice question! $\endgroup$
    – Jason Gaitonde
    Nov 21, 2020 at 16:34

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