We present some classes of graphs which satisfy the acyclic edge colouring conjecture which states that any graph can be acyclically edge coloured with at most ∆ + 2 colours.

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer’s conditions. One of these, which we call Clique...

An acyclic coloring of a graph G is a coloring of the vertices of G, where no two adjacent vertices of G receive the same color and no cycle of G is bichromatic. An acyclic k-coloring of G is an acyclic coloring of G using at most k colors. In this paper we prove that any triangulated plane graph G with n vertices has a subdivision that is acyclically 4-colorable, where the number of division v...

Expanding on a recent definition by Bialostocki and Voxman, we define the rainbow ramsey number RR(G1, G2) of two graphs G1 and G2 to be the minimum integer N such that any edge-coloring of the complete graph KN with any number of colors must contain either a copy of G1 with every edge the same color or a copy of G2 with every edge a different color. This number is well-defined if G1 is a star ...

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G is said k-choosable. A graph is said to be acyclically k-...

For two graphs S and T , the constrained Ramsey number f(S, T ) is the minimum n such that every edge coloring of the complete graph on n vertices (with any number of colors) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T . Here, a subgraph is said to be rainbow if all of its edges have different colors. It is an immediate consequence of the Erdős-Rado Canoni...

A (1, λ)-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge. A graph G is called α-linear if there exists an integral constant β such that e(G′) ≤ αv(G′) + β for each G′ ⊆ G. In this paper, it is shown that every (1, λ)-embedded graph G is 4-linear for all possible λ, and is acyclicly edge-(3∆(G) + 70)-c...

The acyclic edge colouring problem is extensively studied in graph theory. The corner-stone of this field is a conjecture of Alon et. al.[1] that a′(G) ≤ ∆(G) + 2. In that and subsequent work, a′(G) is typically bounded in terms of ∆(G). Motivated by this we introduce a term gap(G) defined as gap(G) = a′(G) − ∆(G). Alon’s conjecture can be rephrased as gap(G) ≤ 2 for all graphs G. In [5] it was...

Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G′ be the graph obtained from G...

The r-acyclic edge chromatic number of a graph is defined to be the minimum number of colours required to produce an edge colouring of the graph such that adjacent edges receive different colours and every cycle C has at least min(|C|, r) colours. We show that (r − 2)d is asymptotically almost surely (a.a.s.) an upper bound on the r-acyclic edge chromatic number of a random d-regular graph, for...