On Mixed Cages

TL;DR

Upper bounds are obtained by general construction methods and computer searches on a regular mixed graph of given girth with minimum possible order.

Abstract

Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.

Figures (14)

• Figure 1: The Behzad-Chartrand-Wall Graph for degree $3$ and girth $5$ has order $13$.
• Figure 2: The AHM Tree for $r=3$, $z=1$, $g=6$.
• Figure 3: The Möbius ladder of order $8$: the $(1,1,5)$-cage. An AHM tree can be found by using any directed path of length four and the three edges incident with the non-endvertices of the path.
• Figure 4: The graph $\vec{C}_7[K_2]$ of order $14$ with $(r,z,g) = (1,2,7)$.
• Figure 5: A $(2,2,6)$-graph with $5$ undirected $6$-cycles.

Theorems & Definitions (18)

• Conjecture 1: Behzad-Chartrand-Wall, bcw
• Theorem 1: The AHM Bound
• Theorem 2
• proof
• Theorem 3: AHM ahm
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6