Cubic vertices in planar hypohamiltonian graphs

Abstract

Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex‐deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.

Every 4-Connected Graph with Crossing Number 2 is Hamiltonian

“…We obtain a planar hypohamiltonian graph with at most two cubic vertices. This, however, contradicts [17,Theorem 3]. If cr(H) = 1, then we glue two copies of H ′ and are led, in the same way, to a contradiction.…”

“…Thereafter, in Section 4, we give applications of Theorem 1-in particular, we (i) discuss how our results relate to the traceability of 3-connected graphs with few 3-cuts and few crossings, (ii) provide, via a toughness argument, 3-connected non-Hamiltonian and non-traceable graphs with small crossing number and few 3-cuts, (iii) show that every 3-connected graph with at most one crossing and containing at most one 3-cut is Hamiltonian (this extends a result of Thomassen [13] which extends the afore-mentioned result of Tutte [15]), and (iv) comment on hypohamiltonian graphs. The latter includes an extension of a theorem of the second author [17] which extends a theorem of Thomassen [13]. In the last section, we give tabular overviews of certain Hamiltonian properties of 3-connected graphs with few crossings and a small number of 3-cuts.…”

“…(We cannot replace the "4" with a "3", since by another result of Thomassen planar cubic hypohamiltonian graphs exist [12].) The second author strengthened this result in several directions [17], one of which states that every hypohamiltonian graph with crossing number 1 contains at least one cubic vertex. We here present an extension of this theorem, but first need two more definitions.…”

“…Since G has at most one cubic vertex, one of these 3-cuts must be non-trivial. We denote this 3-cut with S. Let H and H ′ be the 3-fragments with ends S. By a lemma of the second author [17], no vertex in S is cubic, so any cubic vertex present in G lies either in V (H) \ S or V (H ′ ) \ S-say the former. Clearly we have cr(H) ≤ 1.…”

“…Since any graph with crossing number 1 can be embedded into the projective plane, it follows that every 4-connected graph with crossing number at most 1 is Hamiltonian-connected. Brinkmann (see [17]) showed that if e and f are the crossing edges in a 4-connected graph G with crossing number 1, then G contains a Hamiltonian cycle avoiding e and f .…”