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.

We introduce the ring of ordinomials, which will be utilized in defining the partial chromatic ordinomials of infinite graphs with certain properties – a generalization of chromatic polynomials of finite graphs.

In algebraic contexts, Weyl group elements are usually represented in terms of generators and relations, where representation is not unique. For computational purposes, a more combinatorial representation for elements of classical Weyl groups as signed permutation vectors was introduced in [5]. This paper characterizes some special classes of Weyl group elements using this notation. These classes are especially useful for the study of symmetric spaces and their representations.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A labeling \( f: V(G) \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E(G) \to \mathbb{Z}_2 \), defined by \( f^*(xy) = f(x) + f(y) \), for each edge \( xy \in E(G) \). For \( i \in \mathbb{Z}_2 \), let \(\text{v}_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}\) and \(\text{e}_f(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.\)A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert \text{v}_f(0) – \text{v}_f(1) \rvert \leq 1.\)The friendly index set of the graph \( G \), \( FI(G) \), is defined as \(\{ \lvert \text{e}_f(0) – \text{e}_f(1) \rvert : \text{the vertex labeling } f \text{ is friendly} \}.\)This is a generalization of graph cordiality. We introduce a graph construction called the root-union and investigate when gaps exist in the friendly index sets of root-unions of stars by cycles.