$K_2$-Hamiltonian Graphs: II

TL;DR

This work advances the theory of $K_2$-hamiltonian graphs by proving that planarity does not save Grünbaum’s conjecture in the $K_2$ setting, and by constructing infinite families of planar and non-planar $K_2$-hypohamiltonian graphs through cell-based and $\Gamma^k$-type frameworks. It couples two constructive methods with extensive computational verification to classify the orders $n$ for which $K_2$-hypohamiltonian graphs exist across general, cubic, planar, and cubic-planar classes, identifying key examples (e.g., the Petersen graph and $G_{48}$), and establishing exponential growth in the planar case. The paper also extends and corrects previous results (Za21), introduces robust tools such as extendable $5$-cycles and dot-product operations to generate larger families, and lays out a suite of open problems on girth constraints and cycle spectra with potential implications for Grünbaum-type questions in graph theory. Overall, the work significantly broadens the catalog of known $K_2$-hypohamiltonian graphs and provides a framework for generating infinite families with verifiable certificates, advancing both theory and computational techniques in Hamiltonicity.

Abstract

In this paper we use theoretical and computational tools to continue our investigation of $K_2$-hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with $K_1$-hamiltonian graphs, that is, graphs in which every vertex-deleted subgraph is hamiltonian. Perhaps surprisingly, there exist graphs that are both $K_1$- and $K_2$-hamiltonian, yet non-hamiltonian, for example, the Petersen graph. Grünbaum conjectured that every planar $K_1$-hamiltonian graph must itself be hamiltonian; Thomassen disproved this conjecture. Here we show that even planar graphs that are both $K_1$- and $K_2$-hamiltonian need not be hamiltonian, and that the number of such graphs grows at least exponentially. Motivated by results of Aldred, McKay, and Wormald, we determine for every integer $n$ that is not 14 or 17 whether there exists a $K_2$-hypohamiltonian, that is, non-hamiltonian and $K_2$-hamiltonian, graph of order $n$, and characterise all orders for which such cubic graphs and such snarks exist. We also describe the smallest cubic planar graph which is $K_2$-hypohamiltonian, as well as the smallest planar $K_2$-hypohamiltonian graph of girth $5$. We conclude with open problems and by correcting two inaccuracies from the first article.

Figures (13)

• Figure 1: The graph $J_{18}$.
• Figure 2: Two depictions of the planar hypohamiltonian $K_2$-hamiltonian graph $G_{48}$ on 48 vertices obtained from three copies of $J_{18}$.
• Figure 3: Graph $A$ (left) and graph $B$ (right) are two non-isomorphic graphs obtained from $G_{52}$ by removing two cubic vertices. The neighbours of these removed vertices are indicated by squares (for the first vertex) and circles (for the second vertex).
• Figure 4: Two $K_2$-hypohamiltonian graphs.
• Figure 5: A planar hypohamiltonian $K_2$-hamiltonian graph of girth 5 on 52 vertices.

Theorems & Definitions (29)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• Lemma 3: Thomassen Th78
• Lemma 4: Za21
• proof : Second proof of Corollary \ref{['cor:infinite_fam']}.