Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

Abstract

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n 0.694 ), and the circumference of a 3-connected claw-free graph is Ω(n 0.121 ).We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m 0.753 ) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n 0.753 ). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs.

Upper Bound Constructions for Untangling Planar Geometric Graphs

“…The shortness exponent of cubic polyhedral graphs is not known. The currently known best lower bound, due to Bilinski et al [2], is λ ≥ x ≈ 0.7532, where x is the real root of 4 1/x − 3 1/x = 2. The currently known best upper bound is λ ≤ log 23 22 ≈ 0.9858 due to Grünbaum and Walther [10].…”

A Relationship Between Thomassen's Conjecture and Bondy's Conjecture

“…Indeed,the proof of Theorem 3.4 also shows that if Conjecture 1.4 is not true, then for any integer k with k ≥ 5, there exist infinitely many essentially 4-edge-connected graphs H with minimum edge degree at least k such that for any closed trail T in H , dom H (T ) = O(|E(H )| α ). Recall that h = |V ≥4 (H)| and H is a counterexample of Conjecture 1.4 .It should be mentioned here that Bilinski et al[3] recently showed that every essentially 3-edge-connected graph H has a closed trail T with dom H (T ) ≥ ( |E(H )|12) β + 2, where β ≈ 0.753; consider the preimage version of Theorem 1.2 in[3].So, if Conjecture 1.4 is not true, then the gap of bounds on max T ((log dom H (T ))/|E(H )|) between the essentially 4-edge-connected case and the essentially 3-edge-connected case would be only the difference between α and β.…”

“…3 , v 4 be four consecutive vertices of C v and let p = pen H (v). We subdivide the edges v 1 v 2 and v 3 v 4 exactly p times and obtain the paths v…”

Circumference of 3-connected cubic graphs

“…This conjecture remained open until a counterexample was found in 1946 by Tutte [11]. There has since been extensive research concerning longest cycles in graphs, see [6] for more references. We use |G| to denote the order of a graph G, i.e., the number of vertices in G; and refer to the length of a longest cycle in G as the circumference of G. We will be concerned with lower bounds on the circumference of 3-connected cubic graphs.Barnette [4] showed that every 3-connected cubic graph of order n has circumference Ω(log n).…”

“…This conjecture was confirmed by Jackson [8], with c = log 2 (1+ √ 5)−1 ≈ 0.694. Bondy and Simonovits [7] constructed an infinite family of 3-connected cubic graphs with circumference Θ(n log 9 8 ) ≈ Θ(n 0.946 ).Recently, Bilinski, Jackson, Ma and Yu [6] showed that every 3-connected cubic graph of order n has circumference Ω(n α ), where α ≈ 0.753 is the real root of 4 1/x − 3 1/x = 2. This is proved by reducing the problem to one about Eulerian subgraphs in 3-edge-connected graphs.In this paper, we further improve this lower bound by considering certain vertex weighted, 2-connected cubic graphs (multiple edges allowed).…”