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Keywords:
graph; vertices; ordering; pseudoordering; upper Hamiltonian number; upper traceable number; upper H-Hamiltonian number; Hamiltonian spectra
graph; vertices; ordering; pseudoordering; upper Hamiltonian number; upper traceable number; upper H-Hamiltonian number; Hamiltonian spectra
Summary:
In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$-Hamiltonian number of a graph $G$. We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs $G$ and $H$ using $H$-Hamiltonian number of $G$. Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper $H$-Hamiltonian number, which is a generalization of the same claim for upper Hamiltonian numbers and upper traceable numbers. Finally we will show that for every connected graph $H$ only paths have maximal $H$-Hamiltonian number.
References:
[1] Dzúrik, M.: Metrické vlastnosti grafů. bachelor thesis (2018).
[2] Okamoto, F., Zhang, P.: On upper traceable numbers of graphs. Math. Bohem. 133 (2008), 389–405. DOI 10.21136/MB.2008.140628 | MR 2472487
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