On List Equitable Total Colorings of the Generalized Theta Graph
Abstract
In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A k-assignment, L, for a graph G assigns a list, L(v), of k available colors to each v ∈ V(G), and an equitable L-coloring of G is a proper coloring, f, of G such that f(v) ∈ L(v) for each v ∈ V(G) and each color class of f has size at most |V(G)|/k . In 2018, Kaul, Mudrock, and Pelsmajer subsequently introduced the List Equitable Total Coloring Conjecture which states that if T is a total graph of some simple graph, then T is equitably k-choosable for each k ≥ \(T), (T)/2 + 2 \ where (T) is the maximum degree of a vertex in T and (T) is the list chromatic number of T. In this paper we verify the List Equitable Total Coloring Conjecture for subdivisions of stars and the generalized theta graph.