Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 ≈ 2.909.

We discuss relationships among T-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral set T that contains 0, the asymptotic T-coloring ratio R(T ) is equal to the fractional chromatic number of the distance graph G(Z, D), where D=T&[0]. This fact is then used to study the distance ...

In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number 1 ≤ 2r+1 2s+1 < og(G), G 2r+1 2s+1 −→ H if and only if G −→ H− 2s+1 2r+1 . Also, for two rational numbers 2r+1 2s+1 < 2p+1 2q+1 and a non-bipartite graph G, we show that G 2r+1 2s+1 < G 2p+1 2q+1 . In the sequel, we introduce an equivalent definition for circular chromat...

The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional ch...

In this note we consider colorings of series-parallel graphs. Specifically, we provide bounds on their fractional and circular chromatic numbers and the defective version of these parameters. The main result is that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1).

Reed conjectured that for every > 0 and ∆ there exists g such that the fractional total chromatic number of a graph with maximum degree ∆ and girth at least g is at most ∆ + 1 + . We prove the conjecture for ∆ = 3 and for even ∆ ≥ 4 in the following stronger form: For each of these values of ∆, there exists g such that the fractional total chromatic number of any graph with maximum degree ∆ and...

An intersection representation of a graph G is a function f : V…G† ! 2 (where S is any set) with the property that uv A E…G† if and only if f …u†V f …v†0h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that p...

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...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.