Part of the
book series (PSNS, volume 16)
An adjacent vertex-distinguishing edge coloring (AVD-coloring) of a graph is a proper edge coloring such that no two neighbors are adjacent to the same set of colors. Zhang et al.  conjectured that every connected graph on at least 6 vertices is AVD (Δ + 2)-colorable, where A is the maximum degree. In this paper, we prove that (Δ + 1) colors are enough when A is sufficiently larger than the maximum average degree, denoted mad. We also provide more precise lower bounds for two graph classes: planar graphs, and graphs with mad < 3. In the first case, Δ ≥ 12 suffices, which generalizes the result of Edwards et al.  on planar bipartite graphs. No other results are known in the case of planar graphs. In the second case, Δ ≥ 4 is enough, which is optimal and completes the results of Wang and Wang  and of Hocquard and Montassier .
Planar Graph Discrete Math Graph Class Edge Coloring Chromatic Index
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