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 $\chi(G)\leq\lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil+1$.
[1]
Landon Rabern.
A Note On Reed's Conjecture
,
2008,
SIAM J. Discret. Math..
[2]
Dieter Gernert,et al.
A COMPUTERIZED SYSTEM FOR GRAPH THEORY, ILLUSTRATED BY PARTIAL PROOFS FOR GRAPH-COLORING PROBLEMS *
,
2008
.
[3]
Ingo Schiermeyer,et al.
Some results on Reed's Conjecture about omega, Delta, and chi with respect to alpha
,
2010,
Discret. Math..
[4]
Andrew D. King.
Claw-free graphs and two conjectures on omega, delta, and chi
,
2009
.
[5]
T. Karthick,et al.
Bounding χ in terms of ω and Δ for some classes of graphs
,
2011,
Discret. Math..
[6]
Frank Harary,et al.
Graph Theory
,
2016
.