Cubic Graphs with Large Circumference Deficit

Abstract

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,

Shortness Coefficient of Cyclically 4-Edge-Connected Cubic Graphs

“…Note that the bound on ρ(C4P) is slightly weaker than what we gave in Theorem 4. The bound on ρ(C4) was due to Máčajová and Mazák [19] which improved a bound by Hägglund who-as Markström wrote in [18, p. 2]-indirectly proved in [13] that ρ(C4) 14 15 . In fact, applying Theorem 7 to the Petersen graph precisely gives us the graph class which was constructed by Máčajová and Mazák: Corollary 9.…”

“…More conservatively, Bondy (see [9]) has conjectured that there is a constant 0 < c < 1 such that the circumference of every n-vertex graph in C4 is at least cn. This would imply ρ(C4) c > 0, while Máčajová and Mazák [19] even conjecture ρ(C4) c 7 8 , and Markström [18] conjectures that ρ(C4) = 0. Despite the lack of non-trivial lower bounds for ρ(C4), an upper bound for ρ(C4) is known: Máčajová and Mazák [19] showed recently that C4 contains an infinite graph family in which the circumference of every n-vertex graph is at most 7 8 n, which implies ρ(C4) 7 8 .…”

“…This would imply ρ(C4) c > 0, while Máčajová and Mazák [19] even conjecture ρ(C4) c 7 8 , and Markström [18] conjectures that ρ(C4) = 0. Despite the lack of non-trivial lower bounds for ρ(C4), an upper bound for ρ(C4) is known: Máčajová and Mazák [19] showed recently that C4 contains an infinite graph family in which the circumference of every n-vertex graph is at most 7 8 n, which implies ρ(C4) 7 8 . Here, we provide a general theorem (Theorem 7) that implies the result of [19].…”

“…Despite the lack of non-trivial lower bounds for ρ(C4), an upper bound for ρ(C4) is known: Máčajová and Mazák [19] showed recently that C4 contains an infinite graph family in which the circumference of every n-vertex graph is at most 7 8 n, which implies ρ(C4) 7 8 . Here, we provide a general theorem (Theorem 7) that implies the result of [19]. We extend our results about planar graphs to the subclass of graphs in C4 that have genus g for any g 0.…”

The smallest nontrivial snarks of oddness 4

“…The well known dominating cycle conjecture [6] implies that every cyclically 4-edgeconnected snark has circumference at least 4n/3, where n is the order of the graph. On the other hand, in [18] Máčajová and Mazák constructed a family of cyclically 4-edgeconnected snarks on 8m vertices with circumference 7m + 2. They also made a conjecture that every cyclically 4-edge-connected cubic has circumference at least 7n/8.…”