Abstract
A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of G is the smallest number of colors in a linear coloring of G. In this paper, we prove that if G is a planar graph without 4-cycles, then lc\({(G)\le \lceil \frac {\Delta}2\rceil+8}\) , where Δ denotes the maximum degree of G.
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Research supported partially by NSFC (No. 11071223, No. 61170302), ZJNSF (No. Z6090150), ZSDZZZZXK08 and IP-OCNS-ZJNU.
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Wang, W., Wang, Y. Linear Coloring of Planar Graphs Without 4-Cycles. Graphs and Combinatorics 29, 1113–1124 (2013). https://doi.org/10.1007/s00373-012-1149-z
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DOI: https://doi.org/10.1007/s00373-012-1149-z