We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.

Let \(G\) be a graph in which each vertex has been coloured using one of \(k\) colours, say \(c_1,c_2,\ldots,c_k\) If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices coloured \(c_i\), \(i = 1,2,\ldots,k\), and \(|n_i – n_j| \leq 1\) for any \(i,j \in \{1,2,\ldots,k\}\), then \(C\) is equitably \(k\)-coloured. An \(m\)-cycle decomposition \(C\) of a graph \(G\) is equitably \(k\)-colourable if the vertices of \(G\) can be coloured so that every \(m\)-cycle in \(C\) is equitably \(k\)-coloured. For \(m = 4,5\) and \(6\), we completely settle the existence problem for equitably \(2\)-colourable \(m\)-cycle decompositions of complete graphs and complete graphs with the edges of a \(1\)-factor removed.

Only the rotational tournament \(U_n\) for odd \(n \geq 5\), has the cycle \(C_n\) as its domination graph. To include an internal chord in \(C_n\), it is necessary for one or more arcs to be added to \(U_n\), in order to create the extended tournament \(U_n^+\). From this, the domination graph of \(U_t\), \(dom(U_n^+)\), may be constructed where \(C_k\), \(3 \leq k \leq n\), is a subgraph of \(dom(U_n^+)\). This paper explores the characteristics of the arcs added to \(U_n\) that are required to create an internal chord in \(C_n\).

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a dominating set if every vertex of \(V(G) – S\) is adjacent to some vertices in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). In this paper, we study the domination number of generalized Petersen graphs \(P(n,3)\) and prove that \(\gamma(P(n,3)) = n – 2\left\lfloor \frac{n}{4} \right\rfloor (n\neq 11)\).

The maximality of the Suzuki group \(\text{Sz}(K,a)\), \(K\) any commutative field of characteristic \(2\) admitting an automorphism \(\sigma\) such that \(x^{\sigma^2} = x^2\), in the symplectic group \(\text{PSp}_4(K)\), is proved.

Let \(k\) be a positive integer and \(G = (V, E)\) be a connected graph of order \(n\). A set \(D \subseteq V\) is called a \(k\)-dominating set of \(G\) if each \(x \in V(G) – D\) is within distance \(k\) from some vertex of \(D\). A connected \(k\)-dominating set is a \(k\)-dominating set that induces a connected subgraph of \(G\). The connected \(k\)-domination number of \(G\), denoted by \(\gamma_k^c(G)\), is the minimum cardinality of a connected \(k\)-dominating set. Let \(\delta\) and \(\Delta\) denote the minimum and the maximum degree of \(G\), respectively. This paper establishes that \(\gamma_k^c(G) \leq \max\{1, n – 2k – \Delta + 2\}\), and \(\gamma_k^c(G) \leq (1 + o_\delta(1))n \frac{ln[m(\delta+1)+2-t]}{m(\delta+1)+2-t}\), where \(m = \lceil \frac{k}{3} \rceil\), \(t = 3 \lceil \frac{k}{3} \rceil – k\), and \(o_\delta(1)\) denotes a function that tends to \(0\) as \(\delta \to \infty\). The later generalizes the result of Caro et al. in [Connected domination and spanning trees with many leaves. SIAM J. Discrete Math. 13 (2000), 202-211] for \(k = 1\).

A graph \(U\) is (induced)-universal for a class of graphs \({X}\) if every member of \({X}\) is contained in \(U\) as an induced subgraph. We study the problem of finding universal graphs with minimum number of vertices for various classes of bipartite graphs: exponential classes, bipartite chain graphs, bipartite permutation graphs, and general bipartite graphs. For exponential classes and general bipartite graphs we present a construction which is asymptotically optimal, while for the other classes our solutions are optimal in order.

The multicolor Ramsey number \(R_r(H)\) is defined to be the smallest integer \(n = n(r)\) with the property that any \(r\)-coloring of the edges of complete graph \(K_n\) must result in a monochromatic subgraph of \(K_n\) isomorphic to \(H\). In this paper, we study the case that \(H\) is a cycle of length \(2k\). If \(2k \geq r+1\) and \(r\) is a prime power, we show that \(R_r(C_{2k}) > {r^2+2k-r-1}\).

The bondage number \(b(D)\) of a digraph \(D\) is the cardinality of a smallest set of arcs whose removal from \(D\) results in a digraph with domination number greater than the domination number of \(D\). In this paper, we present some upper bounds on bondage number for oriented graphs including tournaments, and symmetric planar digraphs.

Let \(G\) be a connected graph. For a vertex \(v \in V(G)\) and an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the representation of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(r(v|\Pi) = (d(v, S_1), d(v, S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is said to be resolving if the \(k\)-vectors \(r(v|\Pi), v \in V(G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is called the partition dimension of \(G\), denoted by \(pd(G)\). A resolving \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\) is said to be connected if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is connected in \(G\). The minimum \(k\) for which there is a connected resolving \(k\)-partition of \(V(G)\) is called the connected partition dimension of \(G\), denoted by \(cpd(G)\). In this paper, the partition dimension as well as the connected partition dimension of the wheel \(W_n\) with \(n\) spokes are considered, by showing that \(\lceil (2n)^{1/3} \rceil \leq pd(W_n) \leq \lceil 2(n)^{1/2} \rceil +1\) and \(cpd(W_n) = \lceil (n+2)/3 \rceil\) for \(n \geq 4\).