On List Equitable Total Colorings of the Generalized Theta Graph - Mudrock Jeffrey A.; Marsh Max; Wagstrom Tim (Discussiones Mathematicae Graph Theory, 2021)
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⌉. Graph G is equitably k-choosable if G is equitably L-colorable whenever L is a k-assignment for G. 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 ≥ max{xℓ(T ), Δ(T )/2 + 2} where Δ(T ) is the maximum degree of a vertex in T and xℓ(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.