Кафедра математики

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A search for regular K3-irregular graphs
(2024) Hak, Artem
For a given graph F, the F-degree of a vertex v in G is the number of subgraphs of G, isomorphic to F, to which v belongs. A graph G is called F-irregular if all vertices of G have distinct F-degrees. In [1], the existence of regular K3-irregular graphs was posed as an open question. Examples of such graphs for regularities r ∈ {10, 11, 12} were constructed in [2]. We analytically prove that no such graphs exist for r ≤ 7, present such a graph for r = 9, and establish bounds on the order for r = 8. We will use t(v) to denote the K3-degree.
Unique eccentric point graphs and their eccentric digraphs
(2023) Hak, Artem; Haponenko, Vladyslav; Kozerenko, Sergiy; Serdiuk, Andrii
We study graph-theoretic properties of eccentric digraphs of unique eccentric point graphs (shortly, uep-graphs). The latter are the connected graphs in which every vertex has a unique eccentric vertex. In particular, we characterize uep-graphs and the corresponding eccentric digraphs in the following classes: self-centered graphs having the number of vertices twice as diameter, block graphs, and graphs with diameter three. Also, we obtain non-trivial properties of weak components in eccentric digraphs of uep-graphs with diameter four and pose several open questions in this direction.