2-connected chordal graphs and line graphs are (1, 5)-choosable. (January 2021) - Wong, Tsai-Lien
Abstract: Suppose G is a graph and k, k ′ are positive integers. A ( k, k ′ ) -list assignment is a mapping L which assigns to each vertex v a set L ( v ) of k real numbers, and assigns to each edge e a set L ( e ) of k ′ real numbers. A proper L -total weighting is a mapping ϕ : V ( G ) ∪ E ( G ) → R such that ϕ ( z ) ∈ L ( z ) for z ∈ V ∪ E, and ∑ e ∈ E ( u ) ϕ ( e ) + ϕ ( u ) ≠ ∑ e ∈ E ( v ) ϕ ( e ) + ϕ ( v ) for every edge u v . A graph G is called ( k, k ′ ) -choosable if for every ( k, k ′ ) -total list assignment L, there is a proper L -total weighting. This paper proves that chordal graphs with no cut-vertex are ( 1, 5 ) -choosable and line graphs are ( 1, 5 ) -choosable.