AVD Total Coloring Conjecture

Establish that for every finite, simple, undirected graph G, there exists an adjacent-vertex-distinguishing total coloring using at most Δ(G)+3 colors. An adjacent-vertex-distinguishing total coloring is a total coloring φ: V(G)∪E(G) → C such that φ restricted to V(G) is a proper vertex coloring, φ restricted to E(G) is a proper edge coloring, φ(uv) ≠ φ(u) for every edge uv ∈ E(G), and for each edge uv, the set of colors on {u}∪{uw : w ∈ N_G(u)} differs from the set of colors on {v}∪{vw : w ∈ N_G(v)}.

Background

Adjacent Vertex Distinguishing (AVD) total coloring strengthens total coloring by requiring that for every edge uv, the color sets seen at u and v (including the vertex color and incident edge colors) are distinct. Zhang et al. proposed the AVD Total Coloring Conjecture, asserting a universal Δ(G)+3 upper bound on the number of colors needed.

This paper proves the conjecture for 3-degenerate graphs. Prior work established the conjecture for several graph classes, including complete graphs, bipartite graphs, outerplanar graphs, 2-degenerate graphs, graphs with Δ(G)=3 and Δ(G)=4, certain planar graphs (with sufficiently large maximum degree), split graphs, and 4-regular graphs, but the conjecture remains open in general.