Optimal frequency assignment and planar list L(2, 1)-labeling

Haiyang Zhu, Junlei Zhu (), Ying Liu, Shuling Wang, Danjun Huang and Lianying Miao
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Haiyang Zhu: Air Force Logistics College
Junlei Zhu: Jiaxing University
Ying Liu: Air Force Logistics College
Shuling Wang: Air Force Logistics College
Danjun Huang: Zhejiang Normal University
Lianying Miao: China University of Mining and Technology

Journal of Combinatorial Optimization, 2022, vol. 44, issue 4, No 31, 2748-2761

Abstract: Abstract G has a list k-L(2, 1)-labeling if for any k-list assignment L, there exists a coloring $$c:V(G)\rightarrow \bigcup \limits _{v\in V} L(v)$$ c : V ( G ) → ⋃ v ∈ V L ( v ) of G such that $$c(v)\in L(v)$$ c ( v ) ∈ L ( v ) for $$\forall v\in V(G)$$ ∀ v ∈ V ( G ) and for $$\forall u,v\in V(G)$$ ∀ u , v ∈ V ( G ) , $$|c(u)-c(v)|\ge 2$$ | c ( u ) - c ( v ) | ≥ 2 if $$d(u,v)=1$$ d ( u , v ) = 1 , $$|c(u)-c(v)|\ge 1$$ | c ( u ) - c ( v ) | ≥ 1 if $$d(u,v)=2$$ d ( u , v ) = 2 . $$\lambda _{2,1}^{l}(G)=\min \{k|G$$ λ 2 , 1 l ( G ) = min { k | G has a list k-L(2, 1)-labeling $$\}$$ } is called the list L(2, 1)-labeling number of G. In this paper, we prove that for planar graph with maximum degree $$\Delta \ge 5$$ Δ ≥ 5 , girth $$g\ge 13$$ g ≥ 13 and without adjacent 13-cycles, $$\lambda _{2,1}^{l}(G)\le \Delta +3$$ λ 2 , 1 l ( G ) ≤ Δ + 3 holds. Moreover, the upper bound $$\Delta +3$$ Δ + 3 is tight.

Keywords: Planar graph; Girth; $$L(2; 1)$$ L ( 2; 1 ) -labeling; List $$L(2; 1)$$ L ( 2; 1 ) -labeling number; 05C10; 05C15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-021-00791-5

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