Further results on the deficiency of graphs (original) (raw)

Discrete Applied Mathematics

A proper t-edge-coloring of a graph G is a mapping α : E(G) → {1,. .. , t} such that all colors are used, and α(e) = α(e) for every pair of adjacent edges e, e ∈ E(G). If α is a proper edge-coloring of a graph G and v ∈ V (G), then the spectrum of a vertex v, denoted by S (v, α), is the set of all colors appearing on edges incident to v. The deficiency of α at vertex v ∈ V (G), denoted by def (v, α), is the minimum number of integers which must be added to S (v, α) to form an interval, and the deficiency def (G, α) of a proper edge-coloring α of G is defined as the sum v∈V (G) def (v, α). The deficiency of a graph G, denoted by def (G), is defined as follows: def (G) = min α def (G, α), where minimum is taken over all possible proper edge-colorings of G. For a graph G, the smallest and the largest values of t for which it has a proper t-edge-coloring α with deficiency def (G, α) = def (G) are denoted by w def (G) and W def (G), respectively. In this paper, we obtain some bounds on w def (G) and W def (G). In particular, we show that for any l ∈ N, there exists a graph G such that def (G) > 0 and W def (G) − w def (G) ≥ l. It is known that for the complete graph K 2n+1 , def (K 2n+1) = n (n ∈ N). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha luszczak posed the following conjecture on the deficiency of nearcomplete graphs: if n ∈ N, then def (K 2n+1 − e) = n − 1. In this paper, we confirm this conjecture.