A connected balanced bipartite graph \(G\) on \(2n\) vertices is almost vertex bipancyclic (i.e., \(G\) has cycles of length \(6, 8, \ldots, 2n\) through each vertex of \(G\)) if it satisfies the following property \(P(n)\): if \(x, y \in V(G)\) and \(d(x, y) = 3\) then \(d(x) + d(y) \geq n + 1\). Furthermore, all graphs except \(C_4\) on \(2n\) (\(n \geq 3\)) vertices satisfying \(P(n)\) are bipancyclic (i.e., there are cycles of length \(4, 6, \ldots, 2n\) in the graph).

Let \(T(m,n)\) denote the number of \(m \times n\) rectangular standard Young tableaux with the property that the difference of any two rows has all entries equal. Let \(T(n) = \sum\limits_{d|n} T(d,n/d)\). We find recurrence relations satisfied by the numbers \(T(m,n)\) and \(\hat{T}(n)\), compute their generating functions, and express them explicitly in some special cases.

A labeling (function) of a graph \(G\) is an assignment \(f\) of nonnegative integers to the vertices of \(G\). Such a labeling of \(G\) induces a labeling of \(L(G)\), the line graph of \(G\), by assigning to each edge \(uv\) of \(G\) the label \(\lvert f(u) – f(v)\rvert\). In this paper we investigate the iteration of such graph labelings.

In this thesis we examine the \(k\)-equitability of certain graphs. We prove the following: The path on \(n\) vertices, \(P_n\), is \(k\)-equitable for any natural number \(k\). The cycle on \(k\) vertices, \(C_n\), is \(k\)-equitable for any natural number \(k\), if and only if all of the following conditions hold:\(n \neq k\); if \(k \equiv 2, 3 \pmod{4}\) then \(n \neq k-1\);if \(k \equiv 2, 3 \pmod{4}\) then \(n \not\equiv k\pmod{2k}\) The only \(2\)-equitable complete graphs are \(K_1\), \(K_2\), and \(K_3\).
The complete graph on \(n\) vertices, \(K_n\), is not \(k\)-equitable for any natural number \(k\) for which \(3 \leq k < n\). If \(k \geq n\), then determining the \(k\)-equitability of \(K_n\) is equivalent to solving a well-known open combinatorial problem involving the notching of a metal bar.The star on \(n+1\) vertices, \(S_n\), is \(k\)-equitable for any natural number \(k\). The complete bipartite graph \(K_{2,n}\) is \(k\)-equitable for any natural number \(k\) if and only if \(n \equiv k-1 \pmod{k}\); or \(n \equiv 0, 1, \ldots, [ k/2 ] – 1 \pmod{k}\);or \(n = \lfloor k/2 \rfloor\) and \(k\) is odd.

The minimal number of triples required to represent all quintuples on an \(n\)-element set is determined for \(n \leq 13\) and all extremal constructions are found. In particular, we establish that there is a unique minimal system on 13 points, namely the 52 collinear triples of the projective plane of order 3.

A set \(T\) with a binary operation \(+\) is called an operation set and denoted as \((T, +)\). An operation set \((S, +)\) is called \(q\)-free if \(qx \notin S\) for all \(x \in S\). Let \(\psi_q(T)\) be the maximum possible cardinality of a \(q\)-free operation subset \((S, +)\) of \((T, +)\).

We obtain an algorithm for finding \(\psi_q({N}_n)\), \(\psi_q({Z}_n)\) and \(\psi_q(D_n)\), \(q \in {N}\), where \({N}_n = \{1, 2, \ldots, n\}\), \(( {Z}_n, +_n)\) is the group of integers under addition modulo \(n\) and \((D_n, +_n)\) is the dihedral group of order \(2n\).