Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F22%3A43964092" target="_blank" >RIV/49777513:23520/22:43964092 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.sciencedirect.com/science/article/pii/S0095895622000077" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0095895622000077</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jctb.2022.01.007" target="_blank" >10.1016/j.jctb.2022.01.007</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs

Popis výsledku v původním jazyce

Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) .

Název v anglickém jazyce

Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs

Popis výsledku anglicky

Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) .

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Combinatorial Theory, Series B

ISSN

0095-8956

e-ISSN

—

Svazek periodika

155

Číslo periodika v rámci svazku

Leden

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

28

Strana od-do

17-44

Kód UT WoS článku

—

EID výsledku v databázi Scopus

2-s2.0-85123885552