Upper bounds for adjacent vertex-distinguishing edge coloring

Junlei Zhu, Yuehua Bu () and Yun Dai
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Junlei Zhu: Zhejiang Normal University
Yuehua Bu: Zhejiang Normal University
Yun Dai: Zhejiang Normal University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 2, No 10, 454-462

Abstract: Abstract An adjacent vertex-distinguishing edge coloring of a graph is a proper edge coloring such that no pair of adjacent vertices meets the same set of colors. The adjacent vertex-distinguishing edge chromatic number is the minimum number of colors required for an adjacent vertex-distinguishing edge coloring, denoted as $$\chi '_{as}(G)$$ χ a s ′ ( G ) . In this paper, we prove that for a connected graph G with maximum degree $$\Delta \ge 3$$ Δ ≥ 3 , $$\chi '_{as}(G)\le 3\Delta -1$$ χ a s ′ ( G ) ≤ 3 Δ - 1 , which proves the previous upper bound. We also prove that for a graph G with maximum degree $$\Delta \ge 458$$ Δ ≥ 458 and minimum degree $$\delta \ge 8\sqrt{\Delta ln \Delta }$$ δ ≥ 8 Δ l n Δ , $$\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }$$ χ a s ′ ( G ) ≤ Δ + 1 + 5 Δ l n Δ .

Keywords: Proper edge coloring; Adjacent vertex-distinguishing edge coloring; Lov $$\acute{a}$$ a ´ sz local lemma (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0187-0

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