Vertex Splitting, Parity Subgraphs and Circuit Covers

Let \(G\) be a \(2\)-edge-connected graph and \(v\) be a vertex of \(G\), and \(F \subset F’ \subset E(v)\) such that \(1 \leq |F|\) and \(|F| + 2 = |F’| \leq d(v) – 1\). Then there is a subset \(F^*\) such that \(F \subset F^* \subset F’\) (here, \(|F^*| = |F| + 1\)), and the graph obtained from \(G\) by splitting the edges of \(F^*\) away from \(v\) remains \(2\)-edge-connected unless \(v\) is a cut-vertex of \(G\). This generalizes a very useful Vertex-Splitting Lemma of Fleischner.
Let \(\mathcal{C}\) be a circuit cover of a bridge-less graph \(G\). The depth of \(\mathcal{C}\) is the smallest integer \(k\) such that every vertex of \(G\) is contained in at most \(k\) circuits of \(\mathcal{C}\). It is conjectured by L. Pyber that every bridge-less graph \(G\) has a circuit cover \(\mathcal{C}\) such that the depth of \(\mathcal{C}\) is at most \(\Delta(G)\). In this paper, we prove that

1. every bridge-less graph \(G\) has a circuit cover \(\mathcal{C}\) such that the depth of \(\mathcal{C}\) is at most \(\Delta(G) + 2\) and
2. if a bridge-less graph \(G\) admits a nowhere-zero \(4\)-flow or contains no subdivision of the Petersen graph, then \(G\) has a circuit cover \(\mathcal{C}\) such that the depth of \(\mathcal{C}\) is at most \(2 \left\lceil 2\Delta(G)/3 \right\rceil\).