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Reed's Conjecture on hole expansions


In 1998, Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G) + \Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we study this conjecture for some {\em expansions} of graphs, that is graphs obtained with the well known operation {\em composition} of graphs.\\ We prove that Reed's Conjecture holds for expansions of bipartite graphs, for expansions of odd holes where the minimum chromatic number of the components is even, when some component of the expansion has chromatic number $1$ or when a component induces a bipartite graph. Moreover, Reed's Conjecture holds if all components have the same chromatic number, if the components have chromatic number at most $4$ and when the odd hole has length $5$. Finally, when $G$ is an odd hole expansion, we prove $\chi(G)\leq\lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil+1$.
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Dates and versions

hal-00694153 , version 1 (03-05-2012)
hal-00694153 , version 2 (29-10-2012)



Jean-Luc Fouquet, Jean-Marie Vanherpe. Reed's Conjecture on hole expansions. 2012. ⟨hal-00694153v2⟩
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