Extremal graphs without three-cycles, four-cycles or five-cycles

Abstract

Given a sea of graphs ψ = {G₁, G₂,..., G _k}, let ex(n; ψ) denote the greatest size of a graph with order n that contains no subgraph isomorphic to some G_ii, 1 ≤ i ≤ k. One of the main classes of problems in extremal graph theory, known as Turán-type problems, is for given n, ψ to determine explicitly the function ex(n; ψ), or to find its asymptotic behavior. Yang Yuansheng et al. investigated the values of ex(n; ψ) for ψ = {C₄}, Garnick et al. investigated them for ψ = {C₃, C₄} and Alabdullatif investigated them for ψ = {C_n-k+1,...,C_n} and ψ = {P_n-k+1,...,P_n}, (1 ≤ k ≤ n - 2). Some bounds for ψ = {C₃, C₄ C₅} were given by Dong and Koh. This paper investigates the values of ex(n; ψ) for ψ = {C₃, C₄, C₅}, n ≤ 42.