Abstract
AbstractWe consider two different notions of graph colouring, namely, the t-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.
Publisher
Springer Science and Business Media LLC
Reference37 articles.
1. Appel, K., Haken, W.: The solution of the four-color-map problem. Sci. Am. 237(4), 108–121 (1977)
2. Arrigo, F., Higham, D., Noferini, V.: Beyond non-backtracking: non-cycling network centrality measures. Proc. Math. Phys. Eng. Scie. 476, 1 (2020)
3. Arrigo, F., Higham, D., Noferini, V., Wood, R.: Weighted enumeration of nonbacktracking walks on weighted graphs. Preprint arXiv:2202.02888 (2022)
4. Backhausz, Á., Szegedy, B., Virág, B.: Ramanujan graphings and correlation decay in local algorithms. Rand. Struct. Algorithms 47(3), 424–435 (2015)
5. Bass, H.: The Ihara–Selberg zeta function of a tree lattice. Int. J. Math. 3(06), 717–797 (1992)