Abstract
A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).
Highlights
We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors
We provide examples showing that eight colors may be necessary
The notions of strong parity vertex coloring and the strong parity chromatic number were defined by Czap and Jendrol’ [3]
Summary
The notions of strong parity vertex coloring and the strong parity chromatic number were defined by Czap and Jendrol’ [3]. Let us recall their definition in an equivalent form. (Throughout the paper, graphs are allowed to have parallel edges but no loops.) Consider a (possibly improper) vertex coloring of G. The restriction to 2-connected graphs in the definition of χs is essential, since there are plane graphs of connectivity one that do not admit any spv-coloring (an example of Czap and Jendrol’ [3] consists of two triangles sharing one vertex). We write FG(v) or dG(v) if this graph is G
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