Abstract
A vertex coloring of a graph G is linear if the subgraph induced by the vertices of any two color classes is the union of vertex-disjoint paths. In this paper, we study the linear coloring of graphs with small girth and prove that: (1) Every planar graph with maximum degree Δ≥39 and girth g≥6 is linearly (⌈Δ2⌉+1)-colorable. (2) There exists an integer Δ0 such that every planar graph with maximum degree Δ≥Δ0 and girth g≥5 is linearly (⌈Δ2⌉+1)-colorable. The latter result is best possible in some sense.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
