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.
Abstract:We determine the exact value of the circular chromatic index of generalized Blanuša snarks of type 1 introduced by Watkins more than two decades ago. In this case, the index takes infinitely many values and can get arbitrarily close to 3. Generalized Blanuša snarks are the first explicit class with this property; until now only finitely many values of the circular chromatic index of snarks have been known.
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.
Abstract:The circumference c(G) of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5-, and 6-edge-connected cubic graphs with circumference ratio c(G)/|V (G)| bounded from above by 0.876, 0.960, and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 4-edge-connected cubic graph is at least 0.75. Up to our knowledge, no upper bounds on this ratio have been known before for cubic graphs with cyclic edge-connectivity above 3. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hägglund and K. Markström, On stable cycles and cycle double covers of graphs with large circumference,
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