Naslov
A Note on the Locally Irregular Edge Colorings of Cacti

Autori
Sedlar, Jelena ; Škrekovski, Riste

Izvornik
Discrete Mathematics Letters (2664-2557) 11 (2023); 1-6

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Locally irregular edge coloring ; Local Irregularity Conjecture ; Cactus graph

Sažetak
A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index χ′irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper, it is proved that χ′irr(G) ≤ 4 for all colorable cactus graphs.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA