We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2− ε)∆t for graphs of maximum degree at most ∆, where ε is some absolute positive constant independent of t. The o...

It is well known that if G is a multigraph then χ(G) ≥ χ(G) := max{∆(G), Γ(G)}, where χ(G) is the chromatic index of G, χ(G) is the fractional chromatic index of G, ∆(G) is the maximum degree of G, and Γ(G) = max{2|E(G[U ])|/(|U | − 1) : U ⊆ V (G), |U | ≥ 3, |U | is odd}. The conjecture that χ(G) ≤ max{∆(G) + 1, dΓ(G)e} was made independently by Goldberg (1973), Anderson (1977), and Seymour (19...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a′(G)≤ +2, where = (G) denotes the maximum degree of the graph. Alon e...

We study graphs which are critical with respect to the chromatic index. We relate these to the Overfull Conjecture and we study in particular their construction from regular graphs by subdividing an edge or by splitting a vertex. In this paper, we consider simple graphs (that is graphs which have no loops or multiple edges). An edge-colouring of a graph G is a map 4 : E(G) -+ cp, where cp is a ...

We determine the circular chromatic index of flower snarks, by showing that χc(F3) = 7/2, χ ′ c(F5) = 17/5 and χ ′ c(Fk) = 10/3 for every odd integer k ≥ 7, where Fk denotes the flower snark on 4k vertices.

The purpose of this study was to obtain additional information about the health of the retina (HR) by measuring the rate of loss of chromatic sensitivity with decreasing light level. The HR(index) is introduced to separate the effects of normal aging from early stage disease. For normal subjects the HR(index is largely independent of age (r(2)~0.1), but ~11% of clinically normal, asymptomatic, ...

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph G, denoted by χ s (G), is the minimum number of colours needed in any strong edge colouring of G. A graph is said to be chordless if there is no cycle in the g...

The strong chromatic index of graphs Mohammad Mahdian Master of Science Graduate Department of C o m p t e r Science University of Toronto 2000 A strong edge-colouring of a graph G is an assignrnent of colours to the edges of G such that every colour class is an induced matching. The minimum number of colours in such a colouring is called the strong chromatic index of G. In 1985, Erdos and Nese...

We conjecture that any graphG with treewidth k and maximum degree ∆(G) ≥ k + √ k satisfies χ′(G) = ∆(G). In support of the conjecture we prove its fractional version.

The Strong Chromatic Index of Halin Graphs By Ziyu Hu A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they have adjacent endpoints. The strong chromatic index of a graph G, denoted by χs(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a planar graph constructed...