Discrete mathematics and Mathematical cybernetics

On some results of the study of $F$-irregular graphs in the class of biconnected graphs $F$

T. S. Dovzhenok, A. V. Filuta, N. E. Chuhai

Secondary school No. 30 of Gomel, 6 Piaсdziasiat gadow BSSR Street, Gomiel 246032, Belarus

Abstract: We consider herein the well-known problem of $F$-irregular graphs in relation to the class of biconnected graphs $F$. It is established that for any natural $n\geq 8$ there exists a $K_{3}$-irregular graph of order $n$. The concept of an almost-almost $F$-irregular graph is introduced, on the basis of which a sufficient condition for the existence of an infinite number of $F$-irregular graphs is found for each graph $F$ from the specified class. It is proved that for any biconnected graph $F$, the minimum of whose vertex degrees is $2$, there are infinitely many $F$-irregular graphs.

Keywords: $F$-degree of a vertex; $F$-irregular graph; biconnected graph; $(K_{3}, K_{2})$ -consistent graph; almost-almost $F$-irregular graph; strong hypothesis about $F$-irregular graphs

UDC: 519.17

Received: 05.10.2023
Revised: 17.06.2024
Accepted: 21.06.2024