A note on Reed's conjecture for triangle-free graphs

被引：0
|
作者：
Abrishami, Gholamreza [1 ]
Erfanian, Ahmad [2 ]
机构：
[1] Ferdowsi Univ Mashhad, Fac Math Sci, Dept Appl Math, POB 1159, Mashhad, Iran
[2] Ferdowsi Univ Mashhad, Ctr Excellence Anal Algebra Struct, Dept Pure Math, POB 1159, Mashhad, Iran
来源：
DISCRETE MATHEMATICS | 2023年 / 346卷 / 12期
关键词：
Reed's conjecture; Graph coloring; Chromatic number; Triangle-free graphs; Girth; CHROMATIC NUMBER; OMEGA; DELTA;
D O I：
10.1016/j.disc.2023.113609
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
Reed's Conjecture states that & chi;(G) & LE; ⠄(⠃(G) + & omega;(G) + 1)/2 ⠅, where & chi; (G), ⠃(G) and & omega;(G) are the chromatic number, maximum degree and clique number of a graph G, respectively. In this note, we prove this conjecture for maximal triangle-free graphs with maximum degree less than 7. Moreover, we show that Reed's Conjecture holds for all graphs with girth at least 5 up to at least 30 vertices and for all triangle-free graphs G up to at least 32 vertices such that & chi;(G) =⠆ 5 which improves similar results given in [Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, J. Graph Theory, 93(2020), 34-48.] & COPY; 2023 Elsevier B.V. All rights reserved.
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