No abstract

Let f be a proper k-coloring of a connected graph G and Π = (V 1 , V 2 , . . . , V k ) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered kIf distinct vertices have distinct color codes, then f is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ L (G). In this paper, we study the locating chromatic number of the join of graphs. We show that when G 1 and G 2 are two connected graphs with diameter at most two, then χwhere G 1 + G 2 is the join of G 1 and G 2 . Also, we determine the locating chromatic numbers of the join of paths, cycles and complete multipartite graphs.

Due to their remarkable application in many branches of applied mathematics such as combinatorics, coding theory, and cryptography, Vandermonde matrices have received a great amount of attention. Maximum distance separable (MDS) codes introduce MDS matrices which not only have applications in coding theory but also are of great importance in the design of block ciphers. Lacan and Fimes introduce a method for the construction of an MDS matrix from two Vandermonde matrices in the finite field. In this paper, we first suggest a method that makes an involutory MDS matrix from the Vandermonde matrices. Then we propose another method for the construction of 2 n × 2 n Hadamard MDS matrices in the finite field G F(2 q ). In addition to introducing this method, we present a direct method for the inversion of a special class of 2 n × 2 n Vandermonde matrices.Keywords MDS matrix · Vandermonde matrix · Hadamard matrix · Blockcipher Mathematics Subject Classification (2000) 11T71 · 14G50 · 51E22 · 94B05 · 20H30 · 15A09 Communicated by J. Jedwab. M. Sajadieh (B) · M. Dakhilalian

A local coloring of a graph G is a function c : V (G) → N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ m S , where m S is the number of edges of the induced subgraph S . The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K (n, k) for some values of n and k is determined.

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