A circulant digraph \(G(a_1, a_2, \ldots, a_k)\), where \(0 < a_1 < a_2 < \ldots < a_k < |V(G)| = n\), is the vertex transitive directed graph that has vertices \(i+a_1, i+a_2, \ldots, i+a_k \pmod{n}\) adjacent to each vertex \(i\). We give the necessary and sufficient conditions for \(G(a_1, a_2)\) to be hamiltonian, and we prove that \(G(a, n-a, b)\) is hamiltonian. In addition, we identify the explicit hamiltonian circuits for a few special cases of sparse circulant digraphs.

We find a family of graphs each of which is not Hall \(t\)-chromatic for all \(t \geq 3\), and use this to prove that the same holds for the Kneser graphs \(K_{a,b}\) when \(a/b \geq 3\) and \(b\) is sufficiently large (depending on \(3 – (a/b)\)). We also make some progress on the problem of characterizing the graphs that are Hall \(t\)-chromatic for all \(t\).

The chromatic sum of \(G\), denoted by \(\sum(G)\), is the minimum sum of vertex colors, taken over all proper colorings of \(G\) using natural numbers. In general, finding \(\sum(G)\) is NP-complete. This paper presents polynomial-time algorithms for finding the chromatic sum for unicyclic graphs and for outerplanar graphs.

We enumerate all order ideals of a garland, a partially ordered set which generalizes crowns and fences. Moreover, we give some bijection between the set of such ideals and the set of certain kinds of lattice paths.

In this paper, we consider transformations between posets \(P\) and \(Q\), whose semi bound graphs are the same. Those posets with the same double canonical posets can be transformed into each other by a finite sequence of two kinds of transformations, called \(d\)-additions and \(d\)-deletions.

A paired-dominating set of a graph \(G\) is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\) is the minimum cardinality of a paired-dominating set of \(G\), and is obviously bounded below by the domination number of \(G\). We give a constructive characterization of the trees with equal domination and paired-domination numbers.

A recent series of papers by Anderson and Preece has looked at half-and-half terraces for cyclic groups of odd order, particularly focusing on those terraces which are narcissistic. We give a new direct product construction for half-and-half terraces which allows us to construct a narcissistic terrace for every abelian group of odd order. We also show that infinitely many non-abelian groups have narcissistic terraces.

Using generating functions of the author \(([1], [2])\), we obtain three infinite classes of combinatorial identities involving partitions with “\(n+t\) copies of \(n\)” introduced by the author and G.E. Andrews [3], and lattice paths studied by the author and D.M. Bressoud [4].

In this paper, we find necessary and sufficient conditions for the existence of a \(6\)-cycle system of \(K_n – E(R)\) for every \(2\)-regular, not necessarily spanning subgraph \(R\) of \(K_n\).

It is known that the smallest complete bipartite graph which is not \(3\)-choosable has \(14\) vertices. We show that the extremal configuration is unique.