Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability of Erd\H{o}s and Simonovits shows if a with $n$ vertices has close to the maximal $t_r(n)$ edges, then it being In this paper we determine exactly graphs at least $m$ edges are farthest from $r$-partite, for any $m\ge t_r(n) - \delta_r n^2$. This extends work by Erd\H{o}s, Gy\H{o}ri Simonovits, proves con...

‎Unmixed bipartite graphs have been characterized by Ravindra and‎ ‎Villarreal independently‎. ‎Our aim in this paper is to‎ ‎characterize unmixed $r$-partite graphs under a certain condition‎, ‎which is a generalization of Villarreal's theorem on bipartite‎ ‎graphs‎. ‎Also, we give some examples and counterexamples in relevance to this subject‎.

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we wi...

We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex adjacent to least $1-1/r+o(1)$ proportion other part, then contains $(r-1)$-st power Hamiltonian cycle

Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...

A latin transversal in a square matrix of order n is a set of entries, no two in the same row or column, which are pairwise distinct. A longstanding conjecture of Ryser states that every Latin square with odd order has a latin transversal. Some results on the existence of a large partial latin transversal can be found in [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdős and Spencer [8] p...

Let r ≥ 2 and c > 0. If G is a graph of order n and the largest eigenvalue of its adjacency matrix satisfies μ (G) ≥ (1− 1/r + c)n, then G contains a complete r-partite subgraph with r − 1 parts of size ⌊

The proof of Theorem 1 consists of the following two lemmas. Recall that K r is the complete r-partite graph with p vertices in each class. In other words, K r = Tr(pr), the Turán graph with pr many vertices. It is easy to see that χ(K r ) = r. Lemma 2. For all c, η > 0, n > 8/η, if G is a graph on n vertices with e(G) ≥ (c + η) ( n 2 ) , then G has a subgraph G′ with n′ ≥ 1 2 √ ηn vertices suc...

For a commutative semigroup S with 0, the zero-divisor graph of S denoted by Γ(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study the case where the graph Γ(S) is complete r-partite for a positive integer r. Also we study the commutative semigroups which are finitely colorable.

For integers k > 0 and r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r, d(v)} different colors. The smallest integer k for which a graph G has a conditional (k, r)-coloring is called the rth order conditional chromatic number, denoted by χr(G). For different val...