We explore the maximum possible toughness among graphs with \(n\) vertices and \(m\) edges in the cases in which \(\lceil \frac{3n}{2}\rceil \leq m < 2n\). In these cases, it is shown that the maximum toughness lies in the interval \([\frac{4}{3}, \frac{3}{2}]\). Moreover, if \(\left\lceil\frac{3n}{2}\right\rceil + 2 \leq m < 2n\), then the value \(\frac{3}{2}\) is achieved. However, if \(m \in \left\{\left\lceil\frac{3n}{2}\right\rceil, \left\lceil\frac{3n}{2}\right\rceil + 1\right\}\), then the maximum toughness can be strictly less than \(\frac{3}{2}\). This provides an infinite family of graphs for which the maximum toughness is not half of the maximum connectivity. The values of maximum toughness are computed for all \(1 \leq n \leq 12\), and some open problems are presented.

A set \(S\) of vertices of a graph \(G = (V, E)\) is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). We define the total domination subdivision number \(sd_{\gamma t}(G)\) to be the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. We give upper bounds on the total domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on \(G\) sufficient to imply that \(sd_{\gamma t}(G) \leq 3\). On the other hand, we show that this constant upper bound does not hold for all graphs. Finally, we show that \(1 \leq sd_{\gamma t}(T) \leq 3\) for any tree \(T\), and characterize the caterpillars \(T\) for which \(sd_{\gamma t}(T) = 3\).

We show that for every \(d \geq 2\), the number of spanning trees of a \(d\)-dimensional grid with \(N\) vertices grows like \(C(d)^N\) for some constant \(C(d)\). Moreover, we show that \(C(d) = 2d-\frac{1}{2}-\frac{5}{16d} + O(d^{-2})\) as \(d\) goes to infinity.

An extended 5-cycle system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a collection of edge-disjoint 5-cycles, 2-tadpoles, and loops that partition the edges of the graph \(K_n^+\) whose vertex set is an \(n\)-set \(V\). In this paper, we show that an extended 5-cycle system of order \(n\) exists for all \(n\) except \(n = 2\) and \(3\).

McMorris, Zaslavsky, and Diny give characterizations of upper bound graphs and double bound graphs in terms of edge clique covers, that is, a family of maximal complete subgraphs that covers all edges. Lundgren and Maybee give a characterization of upper bound graphs using a concept of non-maximal complete subgraphs. In this paper, we present characterizations of double bound graphs and semi-bound graphs in terms of edge covers of non-maximal complete subgraphs.

We consider families of linear self-orthogonal and self-dual codes over the ring \({Z}_4\), which are generated by weighing matrices \(W(n, k)\) with \(k \equiv 0 \pmod{4}\), whose entries are interpreted as elements of the ring \({Z}_4\). We obtain binary formally self-dual codes of minimal Hamming distance 4 by applying the Gray map to the quaternary codes generated by \(W(n, 4)\).

Let \(G = (V, E)\) be a simple, undirected graph. A set of vertices \(D\) is called an odd dominating set if for every vertex \(v \in V(G)\), \(|N[v] \cap D| \equiv 1 \pmod{2}\). The minimum cardinality of an odd dominating set is called the odd domination number of \(G\). It is well known that every graph contains an odd dominating set, but this parameter has been studied very little. Our aim in this paper is to explore some basic features of the odd domination number and to compare it with the domination number of the graph, denoted by \(\gamma(G)\). In addition, extremal values of \(\gamma_{odd}(G)\) are calculated for several classes of graphs and a Nordhaus-Gaddum type inequality \(\gamma_{odd}(G) + \gamma_{odd}(\overline{G})\) is considered.

In this paper, it will be shown that a Skolem sequence of order \(n \equiv 0,1 \pmod{4}\) implies the existence of a graceful tree on \(2n\) vertices which exhibits a perfect matching or a matching on \(2n-2\) vertices. It will also be shown that a Hooked-Skolem sequence of order \(n \equiv 2,3 \pmod{4}\) implies the existence of a graceful tree on \(2n+1\) vertices which exhibits a matching on either \(2n\) or \(2n-2\) vertices. These results will be established using an algorithmic approach.

For \(k \geq 1\) an integer, a set \(D\) of vertices of a graph \(G = (V, E)\) is a \(k\)-dominating set of \(G\) if every vertex in \(V – D\) is within distance \(k\) from some vertex of \(D\). The \(k\)-domination number \(\gamma_k(G)\) of \(G\) is the minimum cardinality among all \(k\)-dominating sets of \(G\). For \(\ell \geq 2\) an integer, the graph \(G\) is \((\gamma_k, \ell)\)-critical if \(\gamma_k(G) = \ell\) and \(\gamma_k(G – v) = \ell – 1\) for all vertices \(v\) of \(G\). If \(G\) is \((\gamma_k, \ell)\)-critical for some \(\ell\), then \(G\) is also called a \(\gamma_k\)-critical graph. For a vertex \(v\) of \(G\), let \(N_k(v) = \{u \in V – \{v\} | d(u,v) \leq k\}\) and let \(\delta_k(G) = \min\{|N_k(v)|: v \in V\}\) and let \(\Delta_k(G) = \max\{|N_k(v)|: v \in V\}\). It is shown that if \(G\) is a nontrivial connected \(\gamma_k\)-critical graph, then \(\delta_k(G) \geq 2k\). Further, it is established that the number of vertices in a \(\gamma-k\)-critical graph \(G\) is bounded above by \((\Delta_k(G)+1)(\gamma_k(G)-1)+1\) and that \(G\) is a \((\gamma_k, \ell)\)-critical graph if and only if the \(k\)th power of \(G\) is a \((\gamma, \ell)\)-critical graph. It is shown that \((k, \ell)\)-critical graphs of arbitrarily large connectivity exist. Moreover, a graph without isolated vertices is shown to be \(\gamma_k\)-critical if and only if each of its blocks is \(\gamma_k\)-critical. Finally it is established that for an integer \(\ell \geq 2\), every graph is an induced subgraph of some \((\gamma_k, \ell)\)-critical graph. This paper concludes with some partially answered questions and some open problems.

We provide complete lists of starters and Skolem sequences which generate perfect one-factorizations of complete graphs up to order \(32\) for starters and \(36\) for Skolem sequences. The resulting perfect one-factorizations are grouped into isomorphism classes, and further analysis of the results is performed.