$K_2$-Hamiltonian Graphs: II
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929425538351104 |
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| author | Goedgebeur, Jan Renders, Jarne Wiener, Gábor Zamfirescu, Carol T. |
| author_facet | Goedgebeur, Jan Renders, Jarne Wiener, Gábor Zamfirescu, Carol T. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_05262 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $K_2$-Hamiltonian Graphs: II Goedgebeur, Jan Renders, Jarne Wiener, Gábor Zamfirescu, Carol T. Combinatorics Discrete Mathematics 05C45, 05C38, 05C10, 05C85, 05C76 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. |
| title | $K_2$-Hamiltonian Graphs: II |
| topic | Combinatorics Discrete Mathematics 05C45, 05C38, 05C10, 05C85, 05C76 |
| url | https://arxiv.org/abs/2311.05262 |