On the size of special class 1 graphs and (P3;k)-co-critical graphs

Abstract

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index χ′(G) of G is Δ or Δ+1. A graph G is class 1 if χ′(G)=Δ, and class 2 if χ′(G)=Δ+1; G is Δ-critical if it is connected, class 2 and χ′(G−e)<χ′(G) for every e∈E(G). A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least (n(Δ−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 (P3;k)-co-critical graphs, where a non-complete graph G is (P3;k)-co-critical if there exists a k-coloring of E(G) such that G does not contain a monochromatic copy of P3 but every k-coloring of E(G+e) contains a monochromatic copy of P3 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 (P3;k)-co-critical graphs. We prove that if G is a (P3;k)-co-critical graph on n≥k+2 vertices, thene(G)≥k2(n−⌈k2⌉−ε)+(⌈k/2⌉+ε2), where 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.