Acyclic Chromatic Index of Fully Subdivided Graphs and Halin Graphs

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). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using ∆ + 1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using ∆ + 1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a′(G) = 3. In this paper we generalise the bound to ∆ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and ∆(G) ≥ 3, then a′(G) = ∆(G). Consider a graph G = (V,E), with E = E(T ) ∪ E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T . Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree ∆. We prove that,