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.