Minimum 2-distance coloring of planar graphs and channel assignment

Junlei Zhu () and Yuehua Bu ()
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Junlei Zhu: Jiaxing University
Yuehua Bu: Zhejiang Normal University

Journal of Combinatorial Optimization, 2018, vol. 36, issue 1, No 6, 55-64

Abstract: Abstract A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. $$\chi _{2}(G)$$ χ 2 ( G ) =min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree $$\Delta $$ Δ , $$\chi _2(G) \le 7$$ χ 2 ( G ) ≤ 7 if $$\Delta \le 3$$ Δ ≤ 3 , $$\chi _2(G) \le \Delta +5$$ χ 2 ( G ) ≤ Δ + 5 if $$4\le \Delta \le 7$$ 4 ≤ Δ ≤ 7 and $$\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1$$ χ 2 ( G ) ≤ ⌊ 3 Δ 2 ⌋ + 1 if $$\Delta \ge 8$$ Δ ≥ 8 . In this paper, we prove that: (1) If G is a planar graph with maximum degree $$\Delta \le 5$$ Δ ≤ 5 , then $$\chi _{2}(G)\le 20$$ χ 2 ( G ) ≤ 20 ; (2) If G is a planar graph with maximum degree $$\Delta \ge 6$$ Δ ≥ 6 , then $$\chi _{2}(G)\le 5\Delta -7$$ χ 2 ( G ) ≤ 5 Δ - 7 .

Keywords: Planar graph; 2-Distance coloring; Maximum degree (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-018-0285-7

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