Abstract
It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.
Highlights
The chromatic number of a graph G, denoted by χ(G), is the minimum number of colours required to colour the vertex set of G so that no two adjacent vertices are assigned the same colour
In this paper we prove that Conjecture 1 holds for line graphs, which are defined
A multigraph is a graph in which multiple edges are permitted between any pair of vertices – all multigraphs in this paper are loopless
Summary
The chromatic number of a graph G, denoted by χ(G), is the minimum number of colours required to colour the vertex set of G so that no two adjacent vertices are assigned the same colour. In [12], Reed conjectured a bound on the chromatic number of any graph G: Conjecture 1 For any graph G, χ(G) ≤. In the same paper, Reed proved that the conjecture holds if ∆(G) is sufficiently large and ω(G) is sufficiently close to ∆(G). He proved that there exists a positive constant α such that χ(G) ≤ α(ω(G)) + (1 − α)(∆(G) + 1) for all graphs. Wl} such that for every vertex fractional chromatic number of G, written χ∗(G), is the v, Si:v∈Si wi = 1 and smallest c for which G has l i=1 wi. Note that it is always bounded above by the chromatic number. In this paper we prove that Conjecture 1 holds for line graphs, which are defined
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