The formula for Turán number of spanning linear forests

Abstract

Let F be a family of graphs. The Turán number ex(n; F ) is defined to be the maximum number of edges in a graph of order n that is F -free. In 1959, Erdős and Gallai determined the Turán number of M k+1 (a matching of size k + 1) as follows: 2 2 + c , where c = 0 if k is odd and c = 1 otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in [22] as special cases.Moreover, we show that our main theorem can imply the Erdős-Gallai Theorem and also give a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős Matching Conjecture.Mathematics Subject Classification (2010): 05C50, 05C45; 05C90

On Zero-Sum Spanning Trees and Zero-Sum Connectivity

“…The first application is a theorem in which we determine exactly the minimum amount of edges in each colour required in order to force a zero-sum or an almost zero-sum spanning path. This is done using Half(P n ) and a recent deep theorem of Ning and Wang [35] on Turán numbers for linear forests.…”

“…In Figure 2 below, the extremal graphs for ex(n, L(n, k)) given in [35] are depicted. Theorem 5 is the key for calculating the exact Turán number ex(n, Half(P n )), which we will need in order to be able to apply item 1 of the Master Theorem.…”

The maximum number of stars in a graph without linear forest

“…Then G tP 3 (n)−u ′ ∼ = G (t−1)P 3 (n−1). By the induction hypothesis, Finally, we remark that the extremal graphs for ex(n, K 2 , M h ) [5], ex(n, K 2 , F ) (F = tP 3 ) [7], ex(n, S r , P k ) [6], ex(n, K s , F ) (F consists of paths of even order) [13], ex(n, K 2 , L n,k ) [10,8], ex(n, K s , L n,k ), ex(n, K * s,t , L n,k ) [12] are all isomorphic to G F (n). On the other hand, it is known that all the graphs involved in the above enumeration, i.e., K s , S r and K * s,t , have a common property that a shifting operation does not decrease the numbers of their copies in an F -free graph.…”

“…For a class H of graphs, the generalized Turán number ex(n, J, H) is defined as the maximum number of copies of J in an H-free graph of order n. For two positive integers n and k, let L n,k be the class of all linear forests with n vertices and k edges and let Wang and Yang in [10] for n ≥ 3k. Further, Ning and Wang [8]…”

The Generalized Turán Number of Spanning Linear Forests

“…Denote by L n,k the family of all linear forests of order n with k edges. Recently, Ning and Wang [15] determined the exact value of ex(n, L n,k ).…”