The integrity of a graph, \(I(G)\), is given by \(I(G) = \min_{S} (|S| + m(G – S))\) where \(S \subseteq V(G)\) and \(m(G – S)\) is the maximum order of the components of \(G – S\). It is shown that, for arbitrary graph \(G\) and arbitrary integer \(k\), the determination of whether \(I(G) \leq k\) is NP-complete even if \(G\) is restricted to be planar. On the other hand, for every positive integer \(k\) it is decidable in time \(O(n^2)\) whether an arbitrary graph \(G\) of order \(n\) satisfies \(I(G) \leq k\). The set of graphs \(\mathcal{G}_k = \{G | I(G) \leq k\}\) is closed under the minor ordering and by the recent results of Robertson and Seymour the set \(\mathcal{O}_k\) of minimal elements of the complement of \(\mathcal{G}_k\) is finite. The lower bound \(|\mathcal{O}_k| \geq (1.7)^k\) is established for \(k\) large.

It is shown that unlike the chromatic polynomial, which does not characterize unions of non-trivial graphs, the circuit polynomial characterizes the unions of many families of graphs. They include unions of chains, cycles and mixtures of these graphs, also unions of complete graphs. It is also shown that in general, if a Hamiltonian graph is characterized by its circuit polynomial, then so also is the union of the graph with itself.

In this paper, we obtain results on the number of constraints \(m\) for some balanced arrays of strength \(4\) when the parameters \(\mu_2\), \(\mu_3\) assume the values \(1\) and \(0\) respectively. It is shown that the maximum value of \(m\) is \(\mu_1 + 4\), and the existence of such an array is established.

A basis is exhibited for the first homology space of a surface over a field. This basis is found by extending a basis of the boundary cycle space of an embedded graph to the cycle space of the graph.

Let \(C(v)\) denote the least number of quintuples of a \(v\)-set \(V\) with the property that every pair of distinct elements of \(V\) occurs in at least one quintuple. It is shown, for \(v \equiv 3 \text{ or } 11\; \text{modulo} \;20\) and \(v \geq 11\), that \(C(v) = \lceil(v-1)/{4}\rceil\) with the possible exception of \(v \in \{83, 131\}\).

An undirected graph of diameter \(D\) is said to be \(D\)-critical if the addition of any edge decreases its diameter. The structure of \(D\)-critical graphs can be conveniently studied in terms of vertex sequences. Following on earlier results, we establish, in this paper, fundamental properties of \(K\)-edge-connected \(D\)-critical graphs for \(K\geq8\) and \(D\geq7\). In particular, we show that no vertex sequence corresponding to such a graph can contain an “internal” term less than \(3\), and that no two non-adjacent internal terms can exceed \(\text{K}-\lceil{2}\sqrt{\text{K}}\rceil+1\). These properties will be used in forthcoming work to show that every subsequence (except at most one) of length three of the vertex sequence contains exactly \(K+1\) vertices, a result which leads to a complete characterization of edge-maximal vertex sequences.

Lander Conjectured: If \(D\) is a \((\text{v, k}, \lambda)\) difference set in an abelian group G with a cyclic Sylow p-subgroup, then p does not divide \((v, n)\), where \(\text{n} = \text{k} – \lambda \).

In a previous paper, the above conjecture was verified for \(\lambda = 3\) and \(\text{k} \leq 500\), except for \(\text{k} = 228, 282\) and \(444\). These three exceptional values are dealt with in this note, thereby verifying Lander’s conjecture completely for \(\lambda = 3\) and \(\text{k} \leq 500\).