arXiv subjects

Sergiy Kozerenko

Publications and source records attributed to Sergiy Kozerenko.

6 recordsLinked to original sources

arXiv2026

There is no $8$-regular $K_3$-irregular graph

A graph is $K_3$-irregular if its vertices belong to pairwise distinct numbers of triangles. We prove that no $8$-regular $K_3$-irregular graph exists, settling the last unresolved case. Following the initial discovery of such graphs for regularities $r \in \{10,11,12\}$ (Stevanovi'c et al., 2024), our previous work (Hak et al., 2025) showed that no such graphs exist for $r \le 7$, provided the first example for $r=9$, and proved that any $8$-regular candidate must have between $17$ and $22$ vertices. We exclude these possible orders for $r=8$ by combining careful analysis of triangle degrees with integer linear programming techniques. Meanwhile, a recent construction (Zhang, 2026) established that regular $K_3$-irregular graphs do exist for all $r \ge 9$. Together with our results, this establishes that an $r$-regular $K_3$-irregular graph exists if and only if $r\geq 9$.

math.CO↗

arXiv2026

Regular $K_3$-irregular graphs

We address the problem proposed by Chartrand, Erdős and Oellermann (1988) about the existence of regular $K_3$-irregular graphs. We first establish bounds on the $K_3$-degrees of such graphs and use them to prove that there are no such graphs with regularities at most $7$. For the regularity $8$, we narrow down the bounds on the order of such graphs to six possible values. We then present an explicit example of a $9$-regular $K_3$-irregular graph. Finally, we discuss an evolutionary algorithm developed to discover such graphs. Using it, we have found such graphs for consecutive regularities from $9$ to $30$.

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arXiv2026

Transformation Semigroup Perspective on the Magma Monoid

The monoid of all binary operations was first introduced by H. S. Kim and J. Neggers in 2008. Since then, different aspects and applications of this monoid were studied, while several questions about its semigroup-theoretic properties remain unanswered. We employ a transformation semigroup perspective to fully characterize principal left and right ideals, idempotent and regular elements of this monoid, as well as provide precise combinatorial enumerations of them. This approach gives a general framework for most of the existing results on ideals in the magma monoid. We also answer several open questions posed in the 2023 PhD dissertation of A. Rafieipour. Finally, we correct an error regarding the description of the center of the magma monoid from the 2011 paper of H. F. Fayomi.

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arXiv2026

Regular $K_3$-regular graphs

We study graphs that are simultaneously regular with respect to the ordinary vertex degree and regular with respect to the triangle degree, that is, the number of triangles containing a given vertex. We call such graphs regular $K_3$-regular. We investigate the (non-)existence of regular $K_3$-regular graphs with prescribed parameters $(r_2,r_3)$, where $r_2$ is the vertex degree and $r_3$ is the triangle degree. General bounds relating vertex and edge triangle degrees are derived, and non-existence results are established for broad ranges of these parameters. Special attention is paid to Turán graphs, for which we establish uniqueness results for certain parameters. The paper concludes with a summary of admissible parameters and several open problems.

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arXiv2024

The unitary Cayley graph of upper triangular matrix rings

The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph $C_{T_n (\mathbb{F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{F})$ over a finite field $\mathbb{F}$. We show that $C_{T_n (\mathbb{F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{\frac{kn(n-1)}{2}}$ and $|\mathbb{F}|=p^k$. In particular, if $|\mathbb{F}|=2$, then the graph $C_{T_n (\mathbb{F})}$ has $2^{n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{m,m}$, where $m=2^{\frac{n(n-1)}{2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{T_n (\mathbb{F})}$.

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arXiv2021

A note on the triameter of graphs

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number of leaves. We show that in a connected block graph any triametral triple of vertices contains a diametral pair and that any diametral pair of vertices can be extended to a triametral triple. We also present several open problems concerning the interplay between triametral triples, diametral pairs and peripheral vertices in median and distance-hereditary graphs.

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