On bipartite‐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 article, we consider the case where such graphs are bipartite. As main results, we show that in this context the Moore‐like bound is attained in the case of diameter k=3, and that bipartite‐mixed graphs of diameter k≥4 do not exist.

New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed Graphs

“…Besides this general bound given above, researchers are also interested in some particular versions of the problem, namely when the graphs are restricted to a certain class, such as the class of bipartite graphs (which was studied by the authors [3]), planar graphs (see Fellows,Hell,and Seyffarth [6], and Tischenko [22]), maximal planar bipartite graphs (see Dalfó,Huemer,and Salas [4]), vertex-transitive graphs (see Machbeth,Šiagiová,Širáň, and Vetrík [12], andŠiagiová and Vetrík [20]), Cayley graphs ( [12,20] and Vetrík [23]), Cayley graphs of Abelian groups (Dougherty and Faber [5]), or circulant graphs (Wong and Coppersmith [24], and Monakhova [15]). In this paper, we are concerned with mixed Abelian Cayley graphs.…”

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

“…This problem has been considered for different families of graphs. For instance: bipartite graphs in Dalfó, Fiol, and López [3]; planar graphs in Fellows, Hell, and Seyfarth [7] and in Tischenko [20]; vertex-transitive graphs in Machbeth, Šiagiová, Širáň, and Vetrík [13], and in Šiagiová and Vetrík [18]; Cayley graphs also in [13,18], and in Vetrík [21]; Cayley graphs of Abelian groups in Dougherty and Faber [6], and Aguiló, Fiol and Pérez [1]; and circulant graphs in Wong and Coppersmith [22], Morillo, Fiol, and Fàbrega [16], Fiol, Yebra, Alegre, and Valero [10], and in Monakhova [15]. For more information, see the comprehensive survey of Miller and Širáň [14].…”

A new general family of mixed graphs

“…Most of the main results can be found in the comprehensive survey of Miller andŠiráň [11]. Two recent contributions on mixed graphs are due to the author, Fiol, and López, which study the sequence mixed graphs [5] and the bipartite mixed graphs [6].…”