A graph with chromatic number k is called k-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 23 vertices and all triangle-free 5-chromatic graphs on 24 vertices with maximum degree at most 7. This implies that Reed’s conject...

In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most 5, and for the pth chromatic number χp, from which follows in...

As proved by Kahn, the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph G we have χ(G) ≤ (1 + o(1))χf (G). We extend this result to quasi-line graphs, an important subclass of claw-free graphs. Furthermore we prove that we can construct a colouring that achieves this bound in polynomial time, giving us an asymptotic approximation...

In 1998, Reed conjectured that for any graph G, χ(G) ≤ ⌈ 2 ⌉, where χ(G), ω(G), and ∆(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we study this conjecture for some expansions of graphs, that is graphs obtained with the well known operation composition of graphs. We prove that Reed’s Conjecture holds for expansions of bipartite graph...

Zykov designed one of the oldest known family of triangle-free graphs with arbitrarily high chromatic number. We determine the fractional chromatic number of the Zykov product of a family of graphs. As a corollary, we deduce that the fractional chromatic numbers of the Zykov graphs satisfy the same recurrence relation as those of the Mycielski graphs, that is an+1 = an + 1 an . This solves a co...