We find the set of integers \(v\) for which \(\lambda K_v\) may be decomposed into sets of four triples forming Pasch configurations, for all \(\lambda\). We also remove the remaining exceptional values of \(v\) for decomposing \(K_v\) into sets of other four-triple configurations.

In this paper, we consider some combinatorial structures called balanced arrays (\(B\)-arrays) with a finite number of elements, and we derive some necessary conditions in the form of inequalities for the existence of these arrays. The results obtained here make use of the Holder Inequality.

We present efficient algorithms for computing the matching polynomial and chromatic polynomial of a series-parallel graph in \(O(n^{3})\) and \(O(n^2)\) time respectively. Our algorithm for computing the matching polynomial generalizes and improves the result in \([13]\) from \(O(n^3 \log n)\) time for trees and the chromatic polynomial algorithm improves the result in \([18]\) from \(O(n^4)\) time. The salient features of our results are the following:
Our techniques for computing the graph polynomials can be applied to certain other graph polynomials and other classes of graphs as well. Furthermore, our algorithms can also be parallelized into NC algorithms.

Given positive integers \(p\) and \(q\), a \((p, q)\)-colouring of a graph \(G\) is a mapping \(\theta: V(G) \rightarrow \{1, 2, \ldots, q\}\) such that \(\theta(u) \neq \theta(v)\) for all distinct vertices \(u, v\) in \(G\) whose distance \(d(u, v) \leq p\). The \(p\)th order chromatic number \(\chi^{(p)}(G)\) of \(G\) is the minimum value of \(q\) such that \(G\) admits a \((p, q)\)-colouring. \(G\) is said to be \((p, q)\)-maximally critical if \(\chi^{(p)}(G) = q\) and \(\chi^{(p)}(G + e) > q\) for each edge \(e\) not in \(G\).
In this paper, we study the structure of \((2, q)\)-maximally critical graphs. Some necessary or sufficient conditions for a graph to be \((2, q)\)-maximally critical are obtained. Let \(G\) be a \((2, q)\)-maximally critical graph with colour classes \(V_1, V_2, \ldots, V_q\). We show that if \(|V_1| = |V_2| = \cdots = |V_k| = 1\) and \(|V_{k+1}| = \cdots = |V_q| = h \geq 1\) for some \(k\), where \(1 \leq k \leq q-1\), then \(h \leq h^*\), where

\[h^* = \max \left\{k, \min\{q – 1, 2(q – 1 – k)\}\right\}.\]

Furthermore, for each \(h\) with \(1 \leq h \leq h^*\), we are able to construct a \((2, q)\)-maximally critical connected graph with colour classes \(V_1, V_2, \ldots, V_q\) such that \(|V_1| = |V_2| = \cdots = |V_k| = 1\) and \(|V_{k+1}| = \cdots = |V_q| = h\).

We define the class of \({hereditary \; clique-Helly \; graphs}\) or HCH \({graphs}\). It consists of those graphs, where the cliques of every induced subgraph obey the so-called `Helly-property’, namely, the total intersection of every family of pairwise intersecting cliques is nonempty. Several characterization and an \(O(|V|^2|E|)\) recognition algorithm for HCH graphs \(G = (V, E)\) are given. It is shown that the clique graph of every HCH graph is a HCH graph, and that conversely every HCH graph is the clique graph of some HCH graph. Finally, it is shown that HCH graphs \(G = (V, E)\) have at most \(|E|\) cliques, whence a maximum cardinality clique can be found in time \(O(|V||E|^2)\) in such a HCH graph.

A weight \(w: E(G) \rightarrow \{1, 2\}\) is called a \((1, 2)\)-eulerian weight of graph \(G\) if the total weight of each edge-cut is even. A \((1, 2)\)-eulerian weight \(w\) of \(G\) is called smallest if the total weight \(w\) of \(G\) is minimum. In this note, we prove that if graph \(G\) is \(2\)-connected and simple, and \(w_0\) is a smallest \((1, 2)\)-eulerian weight, then either \(|E_{w_0 = \text{even}}|\leq|V(G)| – 3\) or \(G = K_4\).

In this paper we prove that a \((v, u; \{4\}, 3)\)-IPBD exists when \(v, u \equiv 2\) or \(3 \pmod{4}\) and \(v \geq 3u + 1\), and then solve the problem of the existence of \((v, u; \{4\}, \lambda)\)-IPBD completely, which generalizes the result of \([7]\).

For a wide range of \(p\), we show that almost every graph \(G\epsilon\mathcal{G}(n,p)\) has no perfect dominating set and for almost every graph \(G\epsilon\mathcal{G}(n,p)\) we bound the cardinality of a set of vertices which can be perfectly dominated. We also show that almost every tree \(T\epsilon\mathcal{T}(n)\) has no perfect dominating set.

Necessary conditions for the existence of group divisible designs with block size three are developed. A computation is described that establishes the sufficiency of these conditions for sixty and fewer elements.

Four \(\{\pm1\}\)-matrices \(A, B, C, D\) of order \(n\) are called good matrices if \(A – I_n\) is skew-symmetric, \(B, C\), and \(D\) are symmetric, \(AA^T + BB^T + CC^T + DD^T = 4nI_n\), and, pairwise, they satisfy \(XY^T = YX^T\). It is known that they exist for odd \(n \leq 31\). We construct four sets of good matrices of order \(33\) and one set for each of the orders \(35\) and \(127\).

Consequently, there exist \(4\)-Williamson type matrices of order \(35\), and a complex Hadamard matrix of order \(70\). Such matrices are constructed here for the first time. We also deduce that there exists a Hadamard matrix of order \(1524\) with maximal excess.