A note on the Ramsey number of even wheels versus stars

For two graphs $G_1$ and $G_2$ the Ramsey number $R(G_1,G_2)$ is the smallest integer $N$, such that for any graph on $N$ vertices either $G$ contains $G_1$ or $\overline{G}$ contains $G_2$. Let $S_n$ be a star of order $n$ and $W_m$ be a wheel of order $m+1$. In this paper, it is shown that $R(W_n,S_n)\leq{5n/2-1}$, where $n\geq{6}$ is even. It was proven a theorem which implies that $R(W_n,S_n)\geq{5n/2-2}$, where $n\geq{6}$ is even. Therefore we conclude that $R(W_n,S_n)=5n/2-2$ or $5n/2-1$, for $n\geq{6}$ and even

On the automorphism group of a possible symmetric (81,16,3)(81,16,3) design

In this paper we study the automorphism group of a possible symmetric $(81,16,3)$ design

Star-critical Ramsey number of K4K_4 versus FnF_n

For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $r$, such that any red/blue coloring of the edges of the graph $K_r$ contains either a red subgraph that is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$. Let $S_k=K_{1,k}$ be a star of order $k+1$ and $K_n\sqcup S_k$ be a graph obtained from $K_n$ by adding a new vertex $v$ and joining $v$ to $k$ vertices of $K_n$. The star-critical Ramsey number $r_*(G,H)$ is the smallest positive integer $k$ such that any red/blue coloring of the edges of graph $K_{r-1}\sqcup S_k$ contains either a red subgraph that is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$, where $r=r(G,H)$. In this paper, it is shown that $r_*(F_n,K_4)=4n+2$, where $n\geq{4}$.Comment: 11 pages, 3 figure

The Metric Dimension of The Tensor Product of Cliques

Let $G$ be a connected graph and $W=\{ w_1, w_2, \ldots, w_k \} \subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1), d(v,w_2), \ldots, d(v,w_k))$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension and is denoted by $dim(G)$. In this paper, we study the metric dimension of tensor product of cliques and prove some bounds. Then we determine the metric dimension of tensor product of two cliques.Comment: 9 pages, no figur

A Note on Induced Path Decomposition of Graphs

Let $G$ be a graph of order $n$. The path decomposition of $G$ is a set of disjoint paths, say $\mathcal{P}$, which cover all vertices of $G$. If all paths are induced paths in $G$, then we say $\mathcal{P}$ is an induced path decomposition of $G$. Moreover, if every path is of order at least 2, then we say $G$ has an IPD. In this paper, we prove that every connected $r$-regular graph which is not complete graph of odd order admits an IPD. Also we show that every connected bipartite cubic graph of order $n$ admits an IPD of size at most $\frac{n}{3}$. We classify all connected claw-free graphs which admit an IPD.Comment: 5 page

Australasian Journal of Combinatorics 15(1997). 00.31-35 SMALLEST DEFINING SETS FOR 2-(10,5,4) DESIGNS G.B. KHOSROVSHAHI

A set of blocks which is a subset of blocks of only one design is called a defining set of that design. In this paper we determine smallest defining sets of the 21 nonisomorphic 2-(10,5,4) designs. 1

balanced

new technique for constructing pairwis

Minimal defining sets for full 2- ( v, 3, v- 2) designs

A t-(v, k, J\) design D = (X, B) with B = Pk(X) is called a full design. For t = 2, k = 3 and any v, we give minimal defining sets for these designs. For v = 6 and v = 7, smallest defining sets are found. 1

ON THE AUTOMORPHISM GROUP OF A POSSIBLE SYMMETRIC (81, 16, 3) DESIGN

Abstract. In this paper we study the automorphism group of a possible symmetric (81, 16, 3) design. 1