We prove some Schur positivity results for the chromatic symmetric function X G of a (hyper)graph G, using connections to group algebra group. The first such connection works (hyper)forests F: we describe coefficients F in terms eigenvalues product Hermitian idempotents algebra, one factor each edge (a more general formula similar shape holds all chordal graphs). Our main application this techn...

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof use...

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible. Consider a graph G where each edge of G is assigned a colour from {1, ..., k} (this is not necessarily a proper edge colouring). A k-adapted colouring is an assignment of colours from {1, ..., k} to the vertices of G such that there is no edge with th...

We shall use the distance chromatic index defined by the present author in early nineties, cf. [5] or [4] of 1993. The edge distance of two edges in a multigraph M is defined to be their distance in the line graph L(M) of M . Given a positive integer d, define the d+-chromatic index of the multigraph M , denoted by q(d)(M), to be equal to the chromatic number χ of the dth power of the line grap...

Giving a planar graph G, let χl(G) and χ ′′ l (G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if a planar graph G without 6-cycles with chord, then χl(G) ≤ ∆(G) + 1 and χ ′′ l (G) ≤ ∆(G) + 2 where ∆(G) ≥ 6.

Abstract We consider two recent conjectures made by Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index exclusive number of graphs. One conjecture posits an exact relationship first invariants; we show that in fact predicted value may be arbitrarily far from truth either direction. In process establish some new bounds on both...

This paper gives new constructions of k-chromatic critical graphs with high minimum degree and high edge density, and of vertex-critical graphs with high edge density. c © 2002 Elsevier Science B.V. All rights reserved.

let $g$ be a simple connected graph. the edge-wiener index $w_e(g)$ is the sum of all distances between edges in $g$, whereas the hyper edge-wiener index $ww_e(g)$ is defined as {footnotesize $w{w_e}(g) = {frac{1}{2}}{w_e}(g) + {frac{1}{2}} {w_e^{2}}(g)$}, where {footnotesize $ {w_e^{2}}(g)=sumlimits_{left{ {f,g} right}subseteq e(g)} {d_e^2(f,g)}$}. in this paper, we present explicit formula fo...

Zhou, H., The chromatic difference sequence of the Cartesian product of graphs: Part II, Discrete Applied Mathematics 41 (1993) 263-267. This paper is a continuation of our earlier paper under the same title. We prove that the normalized chromatic difference sequences of the Cartesian powers of a Cayley graph on a finite Abelian group are stable; and that if the chromatic difference sequence of...