Chvátal conjectured that if \(G\) is a \(k\)-tough graph and \(k|V(G)|\) is even, then \(G\) has a \(k\)-factor. In \([5\) it was proved that Chvátal’s conjecture is true. Katerinis\([2]\) presented a toughness condition for a graph to have an \([a, b]\)-factor. In this paper, we prove a stronger result: every \((a – 1 + a/b)\)-tough graph satisfying all necessary conditions has an \([a, b]\)-factor containing any given edge and another \([a, b]\)-factor excluding it. We also discuss some special cases of the above result.

R.A. Bailey has conjectured that all finite groups except elementary Abelian \(2\)-groups with more than one factor have \(2\)-sequencings (i.e., terraces). She verified this for all groups of order \(n\), \(n \leq 9\). Results proved since the appearance of Bailey’s paper make it possible to raise this bound to \(n \leq 87\) with \(n = 64\) omitted. Relatively few groups of order not \(2^n\), \(n \in \{4, 5\}\) must be handled by machine computation.

A set \(S\) of vertices of a graph \(G = (V, E)\) is a global dominating set if \(S\) dominates both \(G\) and its complement \(\overline{G}\). The concept of global domination was first introduced by Sampathkumar. In this paper, we extend this notion to irredundancy. A set \(S\) of vertices will be called universal irredundant if \(S\) is irredundant in both \(G\) and \(\overline{G}\). A set \(S\) will be called global irredundant if for every \(x\) in \(S\), \(x\) is an irredundant vertex in \(S\) either in \(G\) or in \(\overline{G}\). We investigate the universal irredundance and global irredundance parameters of a graph. It is also shown that the determination of the upper universal irredundance number of graphs is NP-Complete.

We enumerate by computer algorithms all simple \(t-(t+7, t+1, 2)\) designs for \(1 \leq t \leq 5\), i.e., for all possible \(t\). This enumeration is new for \(t \geq 3\). The number of nonisomorphic designs is equal to \(3, 13, 27, 1\) and \(1\) for \(t = 1, 2, 3, 4\) and \(5\), respectively. We also present some properties of these designs, including orders of their full automorphism groups and resolvability.

Let \(G\) be a finite simple graph. The vertex clique covering number \({vcc}(G)\) of \(G\) is the smallest number of cliques (complete subgraphs) needed to cover the vertex set of \(G\). In this paper, we study the function \({vcc}(G)\) for the case when \(G\) is \(r\)-regular and \((r-2)\)-edge-connected. A sharp upper bound for \({vcc}(G)\) is determined. Further, the set of possible values of \({vcc}(G)\) when \(G\) is a \(4\)-regular connected graph is determined.