Abstract
Let k be a positive integer. An adjacent vertex distinguishing (for short, AVD) totalk-coloring ϕ of a graph G is a proper total k-coloring of G such that no pair of adjacent vertices have the same set of colors, where the set of colors at a vertex v is {ϕ(v)}∪{ϕ(e):e is incident to v}. Zhang et al. conjectured in 2005 that every graph with maximum degree Δ has an AVD total (Δ+3)-coloring. Recently, Papaioannou and Raftopoulou confirmed the conjecture for 4-regular graphs. In this paper, by applying the Combinatorial Nullstellensatz, we verify the conjecture for all graphs with maximum degree 4.
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