Abstract
A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.
Highlights
The graph G can be viewed as the signed graph (G, ∅)
A circuit C of G is balanced if |C ∩ Σ| is even, otherwise it is unbalanced
We say that a subgraph of (G, Σ) is unbalanced if it contains an unbalanced circuit, otherwise it is balanced
Summary
For a positive integer k, we say that a signed graph (G, Σ) has a k-cover if there is a family C of signed circuits of (G, Σ) such that each edge of G belongs to exactly k members of C. For ordinary graphs G (signed graph (G, Σ) with Σ = ∅), a k-cover of G is just a family of circuits which together covers each edge of G exactly k times. Let G be the signed graph obtained from A and B by joining A and B with two internally disjoint paths of length 2m + 1 such that the two paths form an unbalanced circuit. In [3], Cheng, Lu, Luo, and Zhang proved that each signed Eulerian graph with an even number of negative edges has a 2-cover.
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