Facial parity edge colouring of plane pseudographs

Odd edge coloring of graphs

“…A generalization of Theorem 1.1 was successfully applied in solving a problem of facial parity edge colorings in [2], and its improvement in [3]. In this paper, an analogous result to Theorem 1.1 is proved for loopless graphs.…”

“…In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3,2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved.…”

“…It was already remarked in [2] that an odd k-edge-coloring of its reduction readily provides an odd k-edge-coloring of any loopless graph G, since the removed edges between any two adjacent vertices may all adopt one arbitrary color used on the remaining edges between them in red(G). Hence,…”

Improved bound on facial parity edge coloring

“…In [2], Czap, Jendrol', and Kardoš defined the FPE-coloring of plane graphs and proved that 92 colors suffice to color any 2-edge connected plane graph. Recently, Czap et al [3] improved the upper bound. They also showed that if G is 3-edge connected, 12 colors suffice, and in case when G is a 4-edge connected plane graph, it has an FPE-coloring with at most 9 colors.…”

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A facial parity edge coloring of a 2-edge connected plane graph is an edge coloring where no two consecutive edges of a facial walk of any face receive the same color. Additionally, for every face f and every color c either no edge or an odd number of edges incident to f are colored by c. Czap, Jendrol', Kardoš and Soták [3] showed that every 2-edge connected plane graph admits a facial parity edge coloring with at most 20 colors. We improve this bound to 16 colors.

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