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Neighbor product distinguishing total colorings of 2-degenerate graphs

Author

Listed:
  • Enqiang Zhu

    (Guangzhou University)

  • Chanjuan Liu

    (Dalian University of Technology)

  • Jiguo Yu

    (Qufu Normal University)

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).

Suggested Citation

  • Enqiang Zhu & Chanjuan Liu & Jiguo Yu, 2020. "Neighbor product distinguishing total colorings of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 72-76, January.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:1:d:10.1007_s10878-019-00455-5
    DOI: 10.1007/s10878-019-00455-5
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    References listed on IDEAS

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    1. Haiying Wang, 2007. "On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3," Journal of Combinatorial Optimization, Springer, vol. 14(1), pages 87-109, July.
    2. Weifan Wang & Danjun Huang, 2014. "The adjacent vertex distinguishing total coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 379-396, February.
    3. Tong Li & Cunquan Qu & Guanghui Wang & Xiaowei Yu, 2017. "Neighbor product distinguishing total colorings," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 237-253, January.
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