The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

Xiaohan Cheng and Jianliang Wu ()
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Xiaohan Cheng: University of Jinan
Jianliang Wu: Shandong University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 1, No 1, 13 pages

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 or incident 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. An adjacent vertex distinguishing total-k-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 $$\chi ^{\prime \prime }_{a}(G)$$ χ a ″ ( G ) . It is known that $$\chi _{a}^{\prime \prime }(G)\le \Delta (G)+3$$ χ a ″ ( G ) ≤ Δ ( G ) + 3 for any planar graph with $$\Delta (G)\ge 10$$ Δ ( G ) ≥ 10 . In this paper, we consider the list version of this coloring and show that if G is a planar graph with $$\Delta (G)\ge 11$$ Δ ( G ) ≥ 11 , then $${ ch}_{a}^{\prime \prime }(G)\le \Delta (G)+3$$ c h a ″ ( G ) ≤ Δ ( G ) + 3 , where $${ ch}^{\prime \prime }_a(G)$$ c h a ″ ( G ) is the adjacent vertex distinguishing total choosability.

Keywords: Adjacent vertex distinguishing total choosability; Planar graph; Maximum degree (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10878-017-0149-6

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