Circumference of 3-connected cubic graphs

Abstract

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order n is Ω(n 0.694 ). Bilinski et al. improved this lower bound to Ω(n 0.753 ) by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to Ω(n 0.8 ). This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent. *

Dual Circumference and Collinear Sets

“…It is known that any planar graph of maximum degree ∆ can be triangulated so that the resulting triangulation has maximum degree 3∆/2 + 11 [21]. This fact, together with Theorem 1 and the result of Liu, Yu, and Zhang [22], implies the following corollary:…”

“…Stated another way, n-vertex members of the family have circumference O(n α ), for α = log 44 (45) < 0.9941. The current best upper bound of this type is due to Grünbaum and Walther [18] who construct a 24-vertex non-Hamiltonian cubic 3-connected planar graph, resulting in a family of graphs in which n-vertex members have circumference O(n α ) for α = log 23 (22) < 0.9859.…”

On the Circumference of 3-Connected Cubic Triangle-Free Plane Graphs

“…For a lower bound of b(n), Jackson [18] showed that an n-vertex 3-connected cubic graph has circumference Ω(n 0.694 ) for sufficiently large n. Bilinski et al [19] improved this to Ω(n 0.753 ), and then, Liu et al [20] improved to Ω(n 0.8 ). is implies b(n) ≥ Ω(n 0.8 ) for sufficiently large n.…”

Shortness Coefficient of Cyclically 4-Edge-Connected Cubic Graphs

“…This is indeed the case, as we shall prove for the graphs in C4P whose face lengths are at most some constant which is larger than 22 that the shortness coefficient is at most 45 46 . Bondy and Simonovits [2] showed that σ(C3) log 9 8 ≈ 0.946, while Liu, Yu, and Zhang [16] showed that σ(C3) 0.8. Walther [27] proved that σ(C3P) log 27 26 (see also Theorem B in [12]), which solves an open problem by Grünbaum and Motzkin.…”