On the size of special class 1 graphs and $(P_3; k)$-co-critical graphs

Abstract

A well-known theorem of Vizing states that if G is a simple graph with maximum degree ∆, then the chromatic index χfor every e ∈ E(G). A long-standing conjecture of Vizing from 1968 states that every ∆-critical graph on n vertices has at least (n(∆ − 1) + 3)/2 edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, χ ′ (G + e) = χ ′ (G) + 1 for every e ∈ E(G). Such graphs have intimate relation to (P 3 ; k)-co-critical graphs, where a noncomplete graph G is (P 3 ; k)-co-critical if there exists a k-coloring of E(G) such that G does not contain a monochromatic copy of P 3 but every k-coloring of E(G + e) contains a monochromatic copy of P 3 for every e ∈ E(G). We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all (P 3 ; k)-co-critical graphs. We prove that if G is a (Pwhere ε is the remainder of n − ⌈k/2⌉ when divided by 2. This bound is best possible for all k ≥ 1 and n ≥ ⌈3k/2⌉ + 2.