On bipartite (1, 1, k)-mixed graphs

Abstract

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs, in terms of the number of vertices, are presented for small diameters. Moreover, two infinite families of such graphs with diameter k and number of vertices of the order of 2k/2 are proposed, one of them being totally regular (1,1)-mixed graphs. In addition, we present two more infinite families called chordal ring and chordal double ring mixed graphs, which are bipartite and related to tessellations of the plane. Finally, we give an upper bound that improves the Moore bound for bipartite mixed graphs for r = z = 1.

The research of C. Dalfó and M. A. Fiol has been supported by AGAUR from the Catalan Government underproject 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00. Theresearch of M. A. Fiol was also supported by a grant from the Universitat Politècnica de Catalunya with referencesAGRUPS-2022 and AGRUPS-2023

Peer Reviewed

Postprint (published version)