We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph G with oddness ω(G) other than the Petersen graph has at least 5.41 · ω(G) vertices, and for each integer k with 2 ≤ k ≤ 6 we construct an infinite family of cubic graphs with cyclic connectivity k and small oddness ratio |V (G)|/ω(G). In particular, for cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of snarks to 7.5, 13, 25, and 99 from the known values 9, 15, 76, and 118, respectively. In addition, we construct a cyclically 4-connected snark of girth 5 with oddness 4 on 44 vertices, improving the best previous value of 46.

Let G be a regular bipartite graph and X⊆E(G). We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph (G,X), that is a graph G with exactly the edges from X being negative, is not equivalent to (G,∅). In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.

Abstract:We show that for each rational number r such that 4

We show that every bridgeless cubic graph G on n vertices other than the Petersen graph has a 2-factor with at most 2(n−2)/15 circuits of length 5. An infinite family of graphs attains this bound. We also show that G has a 2-factor with at most n/5.83 odd circuits. This improves the previously known bound of n/5.

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This article establishes a relationship between the real (circular) flow number of a graph and its cycle rank. We show that a connected graph with real flow number p/q + 1, where p and q are two relatively prime numbers must have cycle rank at least p + q − 1. A special case of this result yields that the real flow number of a 2-connected cubic graph with chromatic index 4 and order at most 8k + 4 is bounded from below by 4 + 1/k. Using this bound we prove that the real flow number of the Isaacs snark I 2k+1 equals 4 + 1/k, completing the upper bound due to Steffen [Steffen, J Graph Theory 36 (2001), 24-34].

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A graph G is k-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree k or less. We prove that every connected planar graph with average degree d ≥ 2 has a 4-degenerate induced subgraph containing at least (38 − d)/36 of its vertices. This shows that every planar graph of order n has a 4-degenerate induced subgraph of order more than 8/9 · n. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.