Det‐extremal cubic bipartite graphs

Abstract

Let G be a connected k-regular bipartite graph with bipartition V ðGÞ ¼ X [ Y and adjacency matrix A. We say G is det-extremal if per ðAÞ ¼ jdetðAÞj. Det-extremal k-regular bipartite graphs exist only for k ¼ 2 or 3. McCuaig has characterized the det-extremal 3-connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of det-extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det-extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp.

Regular bipartite graphs with all 2-factors isomorphic

“…In [1] the structure of 2-connected cubic det-extremal bipartite graphs is characterized. Part of that characterization is summarized in Theorem 2.4 below.…”

Colouring problems for symmetric configurations with block size 3

“…The third author acknowledges support from the APVV Research Grants 15‐0220 and 17‐0428, and the VEGA Research Grants 1/0142/17 and 1/0238/19. We thank the anonymous referees for pointing out to us the work of Funk et al [14] which completed the existence spectrum of weakly 3‐chromatic configurations.…”

“…All configurations for were enumerated by Betten, Brinkmann and Pisanski [6] in a paper published in 2000, leaving only the values unresolved. The problem was finally solved in 2003 by Funk et al [14]. A bipartite graph with bipartition such that is said to be det‐extremal if its biadjacency matrix satisfies the equation .…”

“…Thomassen [23] pointed out that a symmetric ‐configuration is blocking set free if and only if its Levi graph is det‐extremal. In [14] the following theorem was proved from which it is an immediate corollary that there are no symmetric configurations without a blocking set for .…”

“…Define the connectivity of a symmetric configuration to be the connectivity of its Levi graph. Funk et al [14] present the following operation. Let and be cubic bipartite graphs which are disjoint, and let with neighbour set and with neighbour set .…”