Circuit Covers of Signed Graphs

Abstract

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient q∈R such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most q·|E(G)|. We show that if G is bridgeless, then q≤9, and in the general case q≤11.

Circuit covers of cubic signed graphs

“…Máčajová et al. obtained the following result. Theorem Let be a 2‐edge‐connected signed graph.…”

“…However, the above examples show that some signed graphs do not have a signed-circuit double cover. The shortest signed-circuit cover problem for signed graph have been studied by Máčajová et al [17] and Cheng et al [5]. Before presenting their results, we need some terminology.…”

“…The shortest signed‐circuit cover problem for signed graph have been studied by Máčajová et al. and Cheng et al. .…”

Shorter signed circuit covers of graphs

“…Indeed, we are not aware of any flow‐admissible signed graph with a shortest cycle cover of length exceeding . An example that attains this value is the Petersen graph with five negative edges forming a 5‐cycle .…”

“…Máčajová et al proved that every flow‐admissible signed graph with edges has a signed circuit cover of length at most . Cheng et al improved the bound to for flow‐admissible signed graphs with negative edges in a minimum signature.…”