Abstract
This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).
Highlights
We consider only simple graphs in this paper unless otherwise stated
A plane graph is a particular drawing of a planar graph in the Euclidean plane
We denote the set of faces of a plane graph G by F(G)
Summary
For a graph G, we denote its vertex set, edge set, minimum degree, and maximum degree by V(G), E(G), δ(G), and Δ(G), respectively. Every graph G is edge-(Δ(G) + 1)-choosable. Consider an arbitrary orientation of G. e graph G is A-colorable if, for every f ∈ F(G, A), there is a vertex coloring c: V(G) ⟶ A such that c(x) − c(y) ≠ f(xy) for each directed edge from x to y. Let A be an Abelian group of order at least k and L: V(G) ⟶ 2A be a list assignment of G.
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