Saturation numbers for tPk with k less than 6

Abstract

A graph G is called F - saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F . The saturation number of F , denoted s a t ( n , F ) , is the minimum number of edges in an n -vertex F -saturated graph. Let t P k denote t disjoint copies of a path on k vertices. For t ≥ 2 , k ≥ 3 and n sufficiently large, Chen et al. (2015) obtained bounds on s a t ( n , t P k ) and proposed a conjecture on s a t ( n , t P k ) . In this paper, we improve the lower bound on s a t ( n , t P 3 ) and determine the exact values of s a t ( n , t P 3 ) for t = 4 , n ≥ 3 t + 2 and t = 5 , n ≥ 3 t + 1 . Moreover, we give some counterexamples for the conjecture of Chen et al. for k ∈ { 4 , 5 } .