Almost Hypohamiltonian Graph


An almost hypohamiltonian graph is a nonhamiltonian graph whose vertex-deleted subgraphs are Hamiltonian with exactly one exception. Specifically, a graph G is almost hypohamiltonian if there exists a vertex w such that G-w is nonhamiltonian but G-v is Hamiltonian for all vertices v!=w. The vertex w may be termed the "exceptional vertex" (Goedgebeur and Zamfirescu1 2019, Tsai 2024).

Wiener (2018) gave examples of planar almost hypohamiltonian graphs on 31 and 36 vertices.

Zamfirescu (2015) found a smallest almost hypohamiltonian graph, which has 17 vertices (Zamfirescu 2015, Goedgebeur and Zamfirescu 2019). Goedgebeur and Zamfirescu1 (2019) found a smallest almost hypohamiltonian graph of girth 4, which has 18 vertices.

No planar almost hypohamiltonian graph of girth 3 is known (Goedgebeur and Zamfirescu 2019, Tsai 2014).

The following table summarizes the smallest almost hypohamiltonian graphs satisfying various crietria (Goedgebeur and Zamfirescu 2019).

classgirthsmallest vertex countnumber
almost hypohamiltonian5172
almost hypohamiltonian5181
almost hypohamiltonian4181
cubic almost hypohamiltonian52610
cubic almost hypohamiltonian5282
cubic almost hypohamiltonian4284
planar almost hypohamiltonian5441
planar cubic almost hypohamiltonian5683

See also

Almost Hamiltonian Graph, Hypohamiltonian Graph, Nonhamiltonian Graph, Snark

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Goedgebeur, J. and Zamfirescu, C. T. "On Almost Hypohamiltonian Graphs." Disc. Math. Theor. Comput. Sci. 21, #5, 2019.The House of Graphs. "Almost Hypohamiltonian Graphs.", C.-C. "Small Planar Hypohamiltonian Graphs." 8 Apr 2024., G. "New Constructions of Hypohamiltonian and Hypotraceable Graphs." J. Graph Th. 87, 526-535, 2018.Zamfirescu, C. T. "On Hypohamiltonian and Almost Hypohamiltonian Graphs." J. Graph Th. 79, 63-81, 2015.Zamfirescu, C. T. "Cubic Vertices in Planar Hypohamiltonian Graphs." J. Graph Th. 90, 189-207, 2019.

Cite this as:

Weisstein, Eric W. "Almost Hypohamiltonian Graph." From MathWorld--A Wolfram Web Resource.

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