Neighbor product distinguishing total colorings of 2-degenerate graphs

Enqiang Zhu (), Chanjuan Liu () and Jiguo Yu
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Enqiang Zhu: Guangzhou University
Chanjuan Liu: Dalian University of Technology
Jiguo Yu: Qufu Normal University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 1, No 5, 72-76

Abstract: Abstract A total-k-neighbor product distinguishing-coloring of a graph G is a mapping $$\phi : V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}$$ϕ:V(G)∪E(G)→{1,2,…,k} such that (1) any two adjacent or incident elements in $$V(G)\cup E(G)$$V(G)∪E(G) receive different colors, and (2) for each edge $$uv\in E(G)$$uv∈E(G), $$f_{\phi }(u)\ne f_{\phi }(v)$$fϕ(u)≠fϕ(v), where $$f_{\phi }(x)$$fϕ(x) denotes the product of the colors assigned to a vertex x and its incident edges under $$\phi $$ϕ. The smallest integer k for which such a coloring of G exists is denoted by $$\chi ^{\prime \prime }_{\prod }(G)$$χ∏″(G). In this paper, by using the famous Combinatorial Nullstellensatz, we show that if G is a 2-degenerate graph with maximum degree $$\varDelta (G)$$Δ(G), then $$\chi ^{\prime \prime }_{\prod }(G) \le \max \{\varDelta (G)+2,7\}$$χ∏″(G)≤max{Δ(G)+2,7}. Our results imply the results on $$K_4$$K4-minor free graphs with $$\varDelta (G)\ge 5$$Δ(G)≥5 (Li et al. in J Comb Optim 33:237–253, 2017).

Keywords: Neighbor product distinguishing total coloring; 2-Degenerate graph; Maximum degree (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-019-00455-5

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