The degree set \( \mathcal{D}(G) \) of a graph \( G \) is the set of degrees of its vertices. It has been shown that when the cardinality of \( \mathcal{D}(G) \) is \( 1 \) (i.e., \( G \) is regular) or \( 2 \) (i.e., \( G \) is bi-regular), the balance index set of \( G \) has simple structures. In this work, we determine the balance index sets of unicyclic graphs and subclasses of \( (p, p+1) \) graphs to demonstrate the application of this recent result. In addition, we give an explicit formula for the balance index sets of subclasses of complete tri-bipartite graphs \( G \) (\(|\mathcal{D}(G)| = 3\)). Structural properties regarding the balance index sets of a general graph \( G \) and application examples are also presented.

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). Any edge labeling \( f \) induces a partial vertex labeling \( f^+ : V(G) \to \mathbb{Z}_2 \) assigning \( 0 \) or \( 1 \) to \( f^+(v) \), \( v \) being an element of \( V(G) \), depending on whether there are more \( 0 \)-edges or \( 1 \)-edges incident with \( v \), and no label is given to \( f^+(v) \) otherwise. For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = |\{v \in V(G) : f^+(v) = i\}| \) and \( e_f(i) = |\{e \in E(G) : f(e) = i\}| \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( |e_f(0) – e_f(1)| \leq 1 \). The edge-balance index set of the graph \( G \) is defined as \( \text{EBI}(G) = \{\lvert v_f(0) – v_f(1) \rvert : f \text{ is edge-friendly}\} \). In this paper, we investigate and present results concerning the edge-balance index sets of flux capacitors and \( L \)-products of stars with cycles.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0,1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \( f^*((u, v)) = f(u) \) if and only if \( f(u) = f(v) \) for each edge \( (u, v) \in E(G) \). For \( i \in A \), 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 \( G \) is said to be friendly if \( |\text{v}_f(0) – \text{v}_f(1)| \leq 1 \). The balance index set of the graph \( G \), \( \text{BI}(G) \), is defined as \( \{|\text{e}_f(0) – \text{e}_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). We determine the balance index sets of Halin graphs of stars and double stars.

For a graph \( G \), the expression \( G \overset{v}{\rightarrow} (a_1, \ldots, a_r) \) means that for any \( r \)-coloring of the vertices of \( G \) there exists a monochromatic \( a_i \)-clique in \( G \) for some color \( i \in \{1, \ldots, r\} \). The vertex Folkman numbers are defined as \( F_v(a_1, \ldots, a_r; q) = \text{min}\{|V(G)| : G \overset{v}{\rightarrow} (a_1, \ldots, a_r) \text{ and } K_q \not\subseteq G\} \). Of these, the only Folkman number of the form \( F(\underbrace{2, \ldots, 2}; r – 1) \) which has remained unknown up to this time is \( F_v(2, 2, 2, 2, 2; 4) \).

We show here that \( F_v(2, 2, 2, 2, 2; 4) = 16 \), which is equivalent to saying that the smallest \( 6 \)-chromatic \( K_4 \)-free graph has \( 16 \) vertices. We also show that the sole witnesses of the upper bound \( F_v(2, 2, 2, 2, 2; 4) \leq 16 \) are the two Ramsey \( (4, 4) \)-graphs on \( 16 \) vertices.

We give cyclic constructions for loop designs with block size \( k = 3, 4, \text{ and } 5 \), and all values of \( v \), and we thereby determine the \((v, \lambda)\) spectrum for LDs with these block sizes. For \( k = 3, 5 \) the \((v, \lambda)\) spectrum for LDs is the same as that for cyclic LDs, but this is not true for \( k = 4 \).

A graph is representable modulo \( n \) if its vertices can be assigned distinct labels from \(\{0,1,2,\ldots,n-1\}\) such that the difference of the labels of two vertices is relatively prime to \( n \) if and only if the vertices are adjacent. The representation number \( \text{rep}(G) \) is the smallest \( n \) such that \( G \) has a representation modulo \( n \). In this paper, we determine the representation number and the Prague dimension (also known as the product dimension) of a complete graph minus a disjoint union of paths.

Given a graph \( G \), let \( E \) be the number of edges in \( G \). A \emph{vertex-magic edge labeling} of \( G \), defined by Wallis [12] in 2001, is a one-to-one mapping from the set of edges onto the set \(\{1, 2, \ldots, E\}\) with the property that at any vertex the sum of the labels of all the edges incident to that vertex is the same constant. In 2003, Hartnell and Rall [5] introduced a two-player game based on these labelings, and proved some nice results about winning strategies on graphs that contain vertices of degree one. In this paper, we prove results about winning strategies for certain graphs with cycles where the minimum degree is two.

A vertex labeling \( f: V \to \{0,1\} \) of the simple graph \( G = (V, E) \) induces a partial edge labeling \( f^*: E \to \{0,1\} \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). Let \( v(i) \) and \( e(i) \) be the number of vertices and edges, respectively, that are labeled \( i \), and define the balance index set of \( G \) as \( \{|e(0) – e(1)| : |v(0) – v(1)| \leq 1\} \). In this paper, we determine the balance index sets of generalized wheels, which are the Zykov sum of a cycle with a null graph.

The channel assignment problem is the problem of assigning radio frequencies to transmitters while avoiding interference. This problem can be modeled and examined using graphs and graph colorings. \( L(2,1) \) coloring was first studied by Griggs and Yeh [6] as a model of a variation of the channel assignment problem. A no-hole coloring, introduced in [4], is defined to be an \( L(2,1) \) coloring of a graph which uses all the colors \(\{0,1,\ldots,k\}\) for some integer \(k\). An \( L(2,1) \) coloring is irreducible, introduced in [3], if no vertex labels in the graph can be decreased and yield another \( L(2,1) \) coloring. A graph \(G\) is inh-colorable if there exists an irreducible no-hole coloring on \(G\).

We consider the inh-colorability of bipartite graphs and Cartesian products. We obtain some sufficient conditions for bipartite graphs to be inh-colorable. We also find the optimal inh-coloring for some Cartesian products, including grid graphs and the rook’s graph.

Let \( G \) be a nontrivial connected graph of order \( n \) and \( k \) an integer with \( 2 \leq k \leq n \). For a set \( S \) of \( k \) vertices of \( G \), let \( \kappa(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( V(T_i) \cap V(T_j) = S \) for every pair \( i, j \) of distinct integers with \( 1 \leq i, j \leq \ell \). A collection \( \{T_1, T_2, \ldots, T_\ell\} \) of trees in \( G \) with this property is called a set of internally disjoint trees connecting \( S \). The \( k \)-connectivity \( \kappa_k(G) \) of \( G \) is defined as \( \kappa_k(G) = \text{min}\{\kappa(S)\} \), where the minimum is taken over all \( k \)-element subsets \( S \) of \( V(G) \). Thus \( \kappa_2(G) \) is the connectivity \( \kappa(G) \) of \( G \). In an edge-colored graph \( G \) in which adjacent edges may be colored the same, a tree \( T \) is a rainbow tree in \( G \) if no two edges of \( T \) are colored the same. For each integer \( \ell \) with \( 1 \leq \ell \leq \kappa_k(G) \), a \( (k, \ell) \)-rainbow coloring of \( G \) is an edge coloring of \( G \) (in which adjacent