A well-known theorem of Erdős and Gallai [1] asserts that a graph with no path of length k contains at most 1 2 (k−1)n edges. Recently Győri, Katona and Lemons [2] gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an r-uniform hypergraph containing no Berge path of length k for all values of r and k except for k = r + 1. We settle the remaining case by proving that an r-uniform hypergraph with more than n edges must contain a Berge path of length r + 1.Given a hypergraph H, we denote the vertex and edge sets of H by V (H) and E(H) respectively. Moreover, let e(H) = |E(H)| and n(H) = |V (H)|.

A set W ⊆ V (G) is called a resolving set, if for each pair of distinct vertices y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim M (G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim M (G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two. * a.behtoei@sci.ikiu.ac.ir † a.davoodi@math.iut.ac.ir

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

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In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n, m, where m ≤ n, let S (n, m) be the smallest integer s for which there exists a PBD on n points whose largest block has size m and the sum of its block sizes is equal to s. Also, let S (n, m) be the smallest integer s for which there exists a PBD on n points which has a block of size m and the sum of it block sizes is equal to s. We prove some lower bounds for S (n, m) and S (n, m). Moreover, we apply these bounds to determine the asymptotic behaviour of the sigma clique partition number of the graph K n − K m , Cocktail party graphs and complement of paths and cycles.