Minimum number of disjoint linear forests covering a planar graph

Huijuan Wang (), Lidong Wu (), Weili Wu () and Jianliang Wu ()
Additional contact information
Huijuan Wang: Shandong University
Lidong Wu: University of Texas at Dallas
Weili Wu: University of Texas at Dallas
Jianliang Wu: Shandong University

Journal of Combinatorial Optimization, 2014, vol. 28, issue 1, No 14, 274-287

Abstract: Abstract Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree $$\varDelta \ge 7$$ Δ ≥ 7 . We have proved that the linear arboricity of $$G$$ G is $$\lceil \frac{\varDelta }{2}\rceil $$ ⌈ Δ 2 ⌉ , if for each vertex $$v\in V(G)$$ v ∈ V ( G ) , there are two integers $$i_v,j_v\in \{3,4,5,6,7,8\}$$ i v , j v ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that any two cycles of length $$i_v$$ i v and $$j_v$$ j v , which contain $$v$$ v , are not adjacent. Clearly, if $$i_v=i, j_v=j$$ i v = i , j v = j for each vertex $$v\in V(G)$$ v ∈ V ( G ) , then we can easily get one corollary: for two fixed integers $$i,j\in \{3,4,5,6,7,8\}$$ i , j ∈ { 3 , 4 , 5 , 6 , 7 , 8 } , if there is no adjacent cycles with length $$i$$ i and $$j$$ j in $$G$$ G , then the linear arboricity of $$G$$ G is $$\lceil \frac{\varDelta }{2}\rceil $$ ⌈ Δ 2 ⌉ .

Keywords: Planar graph; Linear forest; Cycle; Covering (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-013-9680-2

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