A tournament \(T = (V, A)\) is \({arc-traceable}\) if for each arc \(xy \in A\), \(xy\) lies on a directed path containing all the vertices of \(V\), i.e., a hamiltonian path. In this paper, we give two extremal results related to arc-traceability in tournaments. First, we show that a non-arc-traceable tournament \(T\) which is \(m\)-arc-strong must have at least \(2^{m+1}+4m-3\) vertices, and we construct an example that shows that this result is best possible. Next, we consider the maximum number of arcs in a strong tournament that are not part of any hamiltonian path. We use the structure of non-arc-traceable tournaments to prove that no strong tournament contains more than \(\frac{n^2-4n+3}{8}\) arcs that are not part of a hamiltonian path, and we give the unique example that shows that this bound is best possible.

We point out that restricted SB triple systems can only exist for \(v \leq 8\). The case \(v = 8\) is especially interesting since it is extremal in that the pair frequencies of the fifteen pairs not involving either \(1\) or \(2\) must be the frequencies \(2, 3, \dots, 16\), in some order.

Let \( a \) and \( b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even,} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd,}
\end{cases}
\]

\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even,} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd.}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short, \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+: V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{ f(u,v) : (u, v) \in E(G) \}.
\]

We investigate \( Q(a)P(b) \) super-edge-graceful labelings for three classes of \( (p,p+1) \)-graphs.

The Ramsey number \( R(C_p, C_q, C_r) \) is the smallest positive integer \( m \) such that no matter how one colors the edges of the \( K_m \) in red, white, and blue, there must be a red \( C_p \), a white \( C_q \), or a blue \( C_r \). In this work, we verified some known \( R(C_p, C_q, C_r) \) values and computed some new \( R(C_p, C_q, C_r) \) values. The results are based on computer algorithms.

A \( (p,q) \) graph \( G \) is total edge-magic if there exists a bijection \( f: V \cup E \to \{1, 2, \ldots, p+q\} \) such that for each \( e = (u,v) \in E \), we have \( f(u) + f(e) + f(v) \) as a constant. For a graph \( G \), denote \( M(G) \) the set of all total edge-magic labelings. The magic strength of \( G \) is the minimum of all constants among all labelings in \( M(G) \), denoted by \( \text{emt}(G) \). The maximum of all constants among \( M(G) \) is called the maximum magic strength of \( G \) and denoted by \( \text{eMt}(G) \).

Hegde and Shetty classify a magic graph as strong if \( \text{emt}(G) = \text{eMt}(G) \), ideal magic if \( 1 \leq \text{eMt}(G) – \text{emt}(G) \leq p \), and \(\textbf{weak magic}\) if \( \text{eMt}(G) – \text{emt}(G) > p \). A total edge-magic graph is called a super edge-magic if \( f(V(G)) = \{1, 2, \ldots, p\} \). The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.

For even codeword length \( n = 2k, k > 1 \) and alphabet size \( \sigma > 1 \), a family of comma-free codes is constructed with \({\left\lfloor \frac{\sigma^2}{3} \right\rfloor}^r \left( \sigma^2 – \left\lfloor \frac{\sigma^2}{3} \right\rfloor \right)^{k-r}\) codewords where \( 1 \leq r < k \). In particular, a new maximal comma-free code with \( n = 4 \) and \( \sigma = 4 \) is given by one of these codes.

If \( K \) is an \( r \)-clique of \( G \) and \( \chi(G) \) decreases by \( r \) upon the removal of all of the vertices in \( K \), then \( K \) is called a critical \( r \)-clique. Two critical cliques are completely independent provided that no vertex in one clique is adjacent to a vertex from the other. An infinite family of graphs is constructed which demonstrates that for every \( s, t \in \mathbb{N} \), there exists a vertex critical graph which admits a critical \( s \)-clique and a critical \( t \)-clique that are completely independent.

In this paper, we obtain a set of inequalities which are necessary conditions for the existence of balanced arrays of strength five, having \( m \) rows (constraints), and with two symbols. We discuss the use of these inequalities to obtain an upper bound on \( m \), and present some illustrative examples.

For a graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), let \( i(G) \) be the number of isolated vertices in \( G \). The \emph{isolated toughness} of \( G \) is defined as \(I(G) = \min\left\{\frac{|S|}{i(G-S)} \mid S \subseteq V(G), i(G-S) \geq 2 \right\},\)if \( G \) is not complete; and \( I(K_n) = n-1 \). In this paper, we investigate the existence of \([a, b]\)-factors in terms of this graph invariant. We proved that if \( G \) is a graph with \( \delta(G) \geq a \) and \( I(G) \geq a \), then \( G \) has a fractional \( a \)-factor. Moreover, if \( \delta(G) \geq a \), \( I(G) > (a-1) + \frac{a-1}{b} \), and \( G-S \) has no \( (a-1) \)-regular component for any subset \( S \) of \( V(G) \), then \( G \) has an \([a, b]\)-factor. The latter result is a generalization of Katerinis’ well-known theorem about \([a, b]\)-factors (P. Katerinis, Toughness of graphs and the existence of factors, \emph{Discrete Math}. 80(1990), 81-92).

We partition the set of spanning trees contained in the complete graph \( K_n \) into spanning trees contained in the complete bipartite graph \( K_{s,t} \). This classification shows that some properties of spanning trees in \( K_n \) can be derived from trees in \( K_{s,t} \). We use Abel’s binomial theorem and the formula for spanning trees in \( K_{s,t} \) to obtain a proof of Cayley’s theorem using partial derivatives. Some results concerning non-isomorphic spanning trees are presented. In particular, we count these trees for \( Q_3 \) and the Petersen graph.