The adjacent vertex distinguishing total chromatic numbers of planar graphs with $$\Delta =10$$ Δ = 10

Xiaohan Cheng (), Guanghui Wang () and Jianliang Wu ()
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Xiaohan Cheng: Shandong University
Guanghui Wang: Shandong University
Jianliang Wu: Shandong University

Journal of Combinatorial Optimization, 2017, vol. 34, issue 2, No 5, 383-397

Abstract: Abstract A (proper) total-k-coloring of a graph G is a mapping $$\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}$$ ϕ : V ( G ) ∪ E ( G ) ↦ { 1 , 2 , … , k } such that any two adjacent elements in $$V (G) \cup E(G)$$ V ( G ) ∪ E ( G ) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. A total-k-adjacent vertex distinguishing-coloring of G is a total-k-coloring of G such that for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$C(u)\ne C(v)$$ C ( u ) ≠ C ( v ) . We denote the smallest value k in such a coloring of G by $$\chi ''_{a}(G)$$ χ a ′ ′ ( G ) . It is known that $$\chi _{a}''(G)\le \Delta (G)+3$$ χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 for any planar graph with $$\Delta (G)\ge 11$$ Δ ( G ) ≥ 11 . In this paper, we show that if G is a planar graph with $$\Delta (G)\ge 10$$ Δ ( G ) ≥ 10 , then $$\chi _{a}''(G)\le \Delta (G)+3$$ χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 . Our approach is based on Combinatorial Nullstellensatz and the discharging method.

Keywords: Adjacent vertex distinguishing total coloring; Planar graph; Maximum degree (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-016-9995-x

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