Generation and New Infinite Families of $K_2$-hypohamiltonian Graphs

TL;DR

This work develops an exhaustive generator for all pairwise non-isomorphic $K_2$-hypohamiltonian graphs of a given order, augmented by new obstructions and bounding criteria to prune the search. It introduces a planarity-preserving amalgam (gluing) operation to build infinite families and uses extendable $5$-cycles to achieve planar $K_2$-hypohamiltonian graphs for all $n\geq 134$, while also constructing a dense infinite family with maximum degree $\frac{n-1}{3}$ and size $2n-5$. The results yield a near-complete order-based characterisation of existence, sharpen lower bounds for planar and bipartite cases, and provide computational and constructive tools that advance understanding of $K_2$-hypohamiltonian graphs and their extremal properties. Together, these contributions blend exhaustive generation with versatile graph-construction techniques to illuminate both the computational landscape and infinite families of $K_2$-hypohamiltonian graphs.

Abstract

We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for $K_2$-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which $K_2$-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite $K_2$-hypohamiltonian graphs. Furthermore, we describe a new operation for creating $K_2$-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar $K_2$-hypohamiltonian graph of order $n$ for every integer $n\geq 134$. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence $K_2$-hypohamiltonian graphs with large maximum degree and size.

Figures (9)

• Figure 1: Visualisation of the amalgam $G$ of $G_1$ and $G_2$.
• Figure 2: All pairwise different traversals of some cycle $\mathfrak{h}$ (in black) through $\{a, a'\}$ (top row) and through $\{b_1, b'_1, b_2, b'_2\}$ (bottom row) up to symmetry. Edges to the left are going to some vertex in $L$, edges to the right are going to some vertex in $R$.
• Figure 3: A visualisation of the amalgam of $G_1$ and $G_2$. Dashed lines represent (part of) the boundary of some face in the embedding. The top image depicts $G_1$, with $a'_1, a_1, b_1, b'_1$ lying on the boundary of the outer face, and $G_2$ after the removal of $a_ib_i$ and $b_ib'_i$. We see that $a'_1, a_1, b_1, b'_1$ still lie on the outer face and that $a'_2, a_2, b_2, b'_2$ now lie on the boundary of some face $F$. In the bottom image we have taken the embedding of $G_2 - a_2b_2 - b_2b'_2$ for which $F$ is the outer face. It is now easily seen that the identification of $a'_1$ with $a'_2$ and $a_1$ with $a_2$ leaves a plane graph and that the addition of the dotted edges $b_1b_2, b'_1b_2$ and $b_1 b'_2$ does as well.
• Figure 4: Planar $K_2$-hypohamiltonian graphs on $50$, $52$, and $53$ vertices. A face for which the boundary is an extendable $5$-cycle is filled in. All vertices for which the vertex-deleted subgraphs are non-hamiltonian are circled. Proofs of these facts are given in GRWZ22.
• Figure 5: Visualisation of a graph with order $n$ from Theorem \ref{['thm:largeMaxDeg']}. A vertex $v_i$ is adjacent to $c_j$ if $i\equiv j\bmod\: 3$.

Theorems & Definitions (35)

• Lemma 1: Aldred, McKay, Wormald AMW97
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• Lemma 4
• proof
• Lemma 5
• proof