Reducing Vizing’s 2-Factor Conjecture to Meredith Extension of Critical Graphs

• Published in: Graphs and combinatorics Vol. 36; no. 5; pp. 1585 - 1591
• Main Authors: Chen, Xiaodong, Ji, Qing, Liu, Mingda
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.09.2020; Springer Nature B.V
• Subjects: Combinatorics; Engineering Design; Graph theory; Graphs; Mathematics; Mathematics and Statistics; Original Paper; 68R10; Grünewald and Steffen construction; 2-Factor; Critical and almost regular graphs; Independence number
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-020-02191-8

A simple graph G is called Δ - critical if χ ′ ( G ) = Δ ( G ) + 1 and χ ′ ( H ) ≤ Δ ( G ) for every proper subgraph H of G , where Δ ( G ) and χ ′ ( G ) are the maximum degree and the chromatic index of G , respectively. Vizing in 1965 conjectured that any Δ -critical graph contains a 2-factor, which is commonly referred to as Vizing’s 2-factor conjecture ; In 1968, he proposed a weaker conjecture that the independence number of any Δ -critical graph with order n is at most n /2,  which is commonly referred to as Vizing’s independence number conjecture . Based on a construction of Δ -critical graphs which is called Meredith extension first given by Meredith, we show that if α ( G ′ ) ≤ ( 1 2 + f ( Δ ) ) | V ( G ′ ) | for every Δ -critical graph G ′ with δ ( G ′ ) = Δ - 1 , then α ( G ) < ( 1 2 + f ( Δ ) ( 2 Δ - 5 ) ) | V ( G ) | for every Δ -critical graph G with maximum degree Δ , where f is a nonnegative function of Δ . We also prove that any Δ -critical graph contains a 2-factor if and only if its Meredith extension contains a 2-factor.