Optimal strong parity edge-coloring of complete graphs

Abstract

A parity walk in an

Facial parity edge colouring

“…One of the motivations for this paper has come from recent papers of Bunde et al [1,2] who introduced parity edge colourings of simple graphs. Studying the parity of the usage of colours along walks suggested two edge-colouring parameters that have interesting properties and applications.…”

“…A parity walk in an edge colouring of a simple graph is a walk along which each colour is used an even number of times. Bunde et al [2] introduced two parameters. Let p(G) be the minimum number of colours in an edge colouring of G having no parity path (a parity edge colouring).…”

“…Since incident edges of the same colour would form a parity path of length 2, every parity edge colouring is a proper edge colouring, and hence p(G) ≥ χ (G), where χ (G) is the chromatic index of G. Since a path is an open walk, no strong parity edge colouring has a parity path. Hence, every strong parity edge colouring is a parity edge colouring and p(G) ≥ p(G) for every graph G. Although there are graphs G with p(G) > p(G) [2], it remains unknown how large p(G) can be when p(G) = k. Elementary results on these parameters appear in [2]. In [1] it is proved that p(K n ) = 2 log(n) − 1 for all n. Moreover, the optimal strong parity edge colouring of the complete n-vertex graph K n is unique when n is a power of 2.…”

“…Hence, every strong parity edge colouring is a parity edge colouring and p(G) ≥ p(G) for every graph G. Although there are graphs G with p(G) > p(G) [2], it remains unknown how large p(G) can be when p(G) = k. Elementary results on these parameters appear in [2]. In [1] it is proved that p(K n ) = 2 log(n) − 1 for all n. Moreover, the optimal strong parity edge colouring of the complete n-vertex graph K n is unique when n is a power of 2. The authors of [2] mentioned that computing p(G) or p(G) is NP-hard even when G is a tree.…”

“…In [1] it is proved that p(K n ) = 2 log(n) − 1 for all n. Moreover, the optimal strong parity edge colouring of the complete n-vertex graph K n is unique when n is a power of 2. The authors of [2] mentioned that computing p(G) or p(G) is NP-hard even when G is a tree. Clearly, the parity edge colouring is such a colouring that each path uses at least one colour an odd number of times.…”