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

Abstract

Mixed graphs can be seen as digraphs with 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 $2^{k/2}$ 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 $r = z = 1$.

Figures (14)

• Figure 1: The only two bipartite Moore $(1,1,3)$-mixed graphs with $8$ vertices.
• Figure 2: Two largest bipartite $(1,1,4)$-mixed graphs (with $12$ vertices, $2$ less than the corresponding bipartite Moore bound).
• Figure 3: The bipartite mixed graph $BDM(2,5)$ and its base graph. The thick lines in $BDM(2,5)$ represent copy $0$ of the lift.
• Figure 4: Five maximal bipartite $(1,1,5)$-mixed graph with $18$ vertices (the corresponding bipartite Moore bound is $24$).
• Figure 5: A bipartite $(1,1,6)$-mixed graph with $30$ vertices (the corresponding bipartite Moore bound is $40$) and its underlying cubic graph. Blue and yellow colors indicate vertex orbits.

Theorems & Definitions (13)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• Lemma 3.4
• proof
• Lemma 4.1
• proof
• Theorem 4.2