Remarks on the Local Irregularity Conjecture

Jelena Sedlar (Avtor), Riste Škrekovski (Avtor)

Povzetek

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by χ$^′_{irr}$(G), is the smallest number of colors used by a locally irregular edge coloring of G. The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χ$^′_{irr}$(B) = 4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.

Ključne besede

locally irregular edge coloring;local irregularity conjecture;unicyclic graph;cactus graph;

Podatki

Jezik:	Angleški jezik
Leto izida:	2021
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UL FMF - Fakulteta za matematiko in fiziko
UDK:	519.17
COBISS:	93453315
ISSN:	2227-7390
Št. ogledov:	101
Št. prenosov:	24
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Slovenski jezik
Sekundarni naslov:	Opombe o domnevi o lokalni iregularnosti
Sekundarne ključne besede:	lokalno iregularno barvanje povezav;domneva o lokalni iregularnosti;uniciklični graf;kaktus graf;
Vrsta dela (COBISS):	Članek v reviji
Strani:	str. 1-10
Letnik:	ǂVol. ǂ9
Zvezek:	ǂiss. ǂ24
Čas izdaje:	2021
DOI:	10.3390/math9243209
ID:	15288102