Conjectured generalizations of Hadwiger’s Conjecture are discussed. Among other results, it is proved that if \(X\) is a set of \(1\), \(2\) or \(3\) vertices in a graph \(G\) that does not have \(K_6\) as a subcontraction, then \(G\) has an induced subgraph that is \(2\)-, \(3\)- or \(4\)-colourable, respectively, and contains \(X\) and at least a quarter, a third or a half, respectively, of the remaining vertices of \(G\). These fractions are best possible.

In 1967 Alspach [1] proved that every arc of a diregular tournament is contained in cycles of all possible lengths. In this paper, we extend this result to bipartite tournaments by showing that every arc of a diregular bipartite tournament is contained in cycles of all possible even lengths, unless it is isomorphic to one of the graphs \(F_{4k} \). Simultaneously, we also prove that an almost diregular bipartite tournament \(R\) is Hamiltonian if and only if \(|V_1| = |V_2|\) and \(R\) is not isomorphic to one of the graphs \(F_{4k-2}\), where \((V_1, V_2)\) is a bipartition of \(R\). Moreover, as a consequence of our first result, it follows that every diregular bipartite tournament of order \(p\) contains at least \(p/4\) distinct Hamiltonian cycles. The graphs \(F_r = (V, A)\), (\(r \geq 2\)) is a family of bipartite tournaments with \(V = \{v_1, v_2, \ldots, v_r\}\) and \(A = \{v_iv_j | j – i \equiv 1 \pmod{4}\}\).

In this paper we study the edge clique graph \(K(G)\) of many classes of intersection graphs \(G\) — such as graphs of boxicity \(\leq k\), chordal graphs and line graphs. We show that in each of these cases, the edge clique graph \(K(G)\) belongs to the same class as \(G\). Also, we show that if \(G\) is a \(W_4\)-free transitivity orientable graph, then \(K(G)\) is a weakly \( \theta \)-perfect graph.

In this paper we construct pairwise balanced designs (PBDs) having block sizes which are prime powers congruent to \(1\) modulo \(5\) together with \(6\). Such a PBD contains \(n = 5r + 1\) points, for some positive integer \(r\). We show that this condition is sufficient for \(n \geq 1201\), with at most \(74\) possible exceptions below this value. As an application, we prove that there exists an almost resolvable BIB design with \(n\) points and block size five whenever \(n \geq 991\), with at most \(26\) possible exceptions below this value.

A Nuclear Design \(ND(v; k, \lambda)\) is a collection \( {B}\) of \(k\)-subsets of a \(v\)-set \(V\), where \( {B} = \mathcal{P}\cap {C} \), where \((V, \mathcal{P})\) is a maximum packing \((PD(v; k,\lambda))\) and \((V, \mathcal{C})\) is a minimum covering \((CD(v; k,\lambda))\) with \(|{B}|\) as large as possible. We construct \(ND(v; 3, 1)\)’s for all \(v\) and \(\lambda\). Along the way we prove that for every leave (excess) possible for \(k = 3\), all \(v,\lambda\), there is a maximum packing (minimum covering) achieving this leave (excess).

A graph \(G\) is defined to be balanced if its average degree is at least as large as the average degree of any of its subgraphs. We obtain a characterization of all balanced graphs with minimum degree one. We prove that maximal \(Q\) graphs are strictly balanced for several hereditary properties \(Q\). We also prove that a graph \(G\) is balanced if and only if its subdivision graph \(S(G)\) is balanced.