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.

We formalize the intuitive question of coloring the bricks of a wall in such a way that no repetition occurs in any row, nor any vertical line intersects two or more bricks with the same color. We achieve a complete classification up to the least number of required colors, among all dimensions of the walls, and all admitted incidences of the bricks. The involved combinatorial structures (namely, \(regular\) \(walls\)) are a special case of more general structures, which can be interpreted as adjacency matrices of suitable directed hypergraphs. Coloring the bricks is equivalent to coloring the arcs of the corresponding hypergraph. Regular walls seem interesting also for their connections with latin rectangles.

Tutte’s \(3\)-flow conjecture is equivalent to the assertion that there exists an orientation of the edges of a \(4\)-edge-connected, \(5\)-regular graph \(G\)for which the out-flow at each vertex is \(+3\) or \(-3\). The existence of one such orientation of the edges implies the existence of an equipartition of the vertices of \(G\) that separates the two possible types of vertices. Such an equipartition is called mod \(3\)-orientable. We give necessary and sufficient conditions for the existence of mod \(3\)-orientable equipartitions in general \(5\)-regular graphs, in terms of:(i) a perfect matching of a bipartite graph derived from the equipartition;(ii) the sizes of cuts in \(G\).Also, we give a polynomial-time algorithm for testing whether an equipartition of a \(5\)-regular graph is mod \(3\)-orientable.