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.

In this paper, we look at generalizations of Stirling numbers which arise for arbitrary integer sequences and their \(k\)-th powers. This can be seen as a complementary strategy to the unified approach suggested in [9]. The investigations of [3] and [14] present a more algebraically oriented approach to generalized Stirling numbers.

In the first and second sections of the paper, we give the corresponding formulas for the generalized Stirling numbers of the second and first kind, respectively. In the third section, we briefly discuss some examples and special cases, and in the last section, we apply the square case to facilitate a counting approach for set partitions of even size.

In this paper, we give two sufficient conditions for a graph to be type \(1\) with respect to the total chromatic number and prove the following results:

(i) If \(G\) and \(H\) are of type \(1\), then \(G \times H\) is of type \(1\);

(ii) If \(\varepsilon(G) \leq v(G) + \frac{3}{2}\Delta(G) – 4\), then \(G\) is of type \(1\).

We prove several results dealing with various counting functions for partitions of an integer into four squares of equal parity. Some are easy consequences of earlier work, but two are new and surprising. That is, we show that the number of partitions of \(72n+ 60\) into four odd squares (distinct or not) is even.

We prove that if \(G\) is a simple graph of order \(n \geq 3k\) such that \(|N(x) \cup N(y)| \geq 3k\) for all nonadjacent pairs of vertices \(x\) and \(y\), then \(G\) contains \(k\) vertex-independent cycles.

The non-planar vertex deletion or vertex deletion \(vd(G)\) of a graph \(G = (V, E)\) is the smallest non-negative integer \(k\) such that the removal of \(k\) vertices from \(G\) produces a planar graph. Hence, the maximum planar induced subgraph of \(G\) has precisely \(|V| – vd(G)\) vertices. The problem of computing vertex deletion is in general very hard; it is NP-complete. In this paper, we compute the non-planar vertex deletion for the family of toroidal graphs \(C_n \times C_m\).

Let \(K_4\backslash e=…\). If we remove the “diagonal” edge, the result is a \(4\)-cycle. Let \((X,B)\) be a \(K_4\backslash e\) design of order \(n\); i.e., an edge-disjoint decomposition of \(K_n\) into copies of \(K_4\backslash e\). Let \(D(B)\) be the collection of “diagonals” removed from the graphs in \(B\) and \(C(B)\) the resulting collection of \(4\)-cycles. If \(C_2(B)\) is a reassembly of these edges into \(4\)-cycles and \(L\) is the collection of edges in \(D(B)\) not used in a \(4\)-cycle of \(C_2(B)\), then \((X, (C_1(B) \cup C_2(B)), L)\) is a packing of \(K_n\) with \(4\)-cycles and is called a metamorphosis of \((X,B)\). We construct, for every \(n = 0\) or \(1\) (mod \(5\)) \(> 6\), \(n \neq 11\), a \(K_4\backslash e\) design of order \(n\) having a metamorphosis into a maximum packing of \(K_n\) with \(4\)-cycles. There exists a maximum packing of \(K_n\) with \(4\)-cycles, but it cannot be obtained from a \(K_4\backslash e\) design.

We investigate the supereulerian graph problems within planar graphs, and we prove that if a \(2\)-edge-connected planar graph \(G\) is at most three edges short of having two edge-disjoint spanning trees, then \(G\) is supereulerian except for a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We also determine several extremal bounds for planar graphs to be supereulerian.

Given an acyclic digraph \(D\), the phylogeny graph \(P(D)\) is defined to be the undirected graph with \(V(D)\) as its vertex set and with adjacencies as follows: two vertices \(x\) and \(y\) are adjacent if one of the arcs \((x,y)\) or \((y,x)\) is present in \(D\), or if there exists another vertex \(z\) such that the arcs \((x,z)\) and \((y,z)\) are both present in \(D\). Phylogeny graphs were introduced by Roberts and Sheng [6] from an idealized model for reconstructing phylogenetic trees in molecular biology, and are closely related to the widely studied competition graphs. The phylogeny number \(p(G)\) for an undirected graph \(G\) is the least number \(r\) such that there exists an acyclic digraph \(D\) on \(|V(G)| + r\) vertices where \(G\) is an induced subgraph of \(P(D)\). We present an elimination procedure for the phylogeny number analogous to the elimination procedure of Kim and Roberts [2] for the competition number arising in the study of competition graphs. We show that our elimination procedure computes the phylogeny number exactly for so-called “kite-free” graphs. The methods employed also provide a simpler proof of Kim and Roberts’ theorem on the exactness of their elimination procedure for the competition number on kite-free graphs.