In this paper, we present new results about the coloring of graphs. We generalize the notion of proper vertex-coloring, introducing the concept of range-coloring of order \( k \). The relation between range-coloring of order \( k \) and total coloring is presented: we show that for any graph \( G \) that has a range-coloring of order \( \Delta(G) \) with \( t \) colors, there is a total coloring of \( G \) that uses \( (t+1) \) colors. This result provides a framework to prove that some families of graphs satisfy the total coloring conjecture. We exemplify with the family of block-cactus graphs.

We show, for \( k = 3, 4, 5 \), that the necessary conditions are sufficient for the existence of graph designs which decompose \( K_v(\lambda, j) \), the complete (multi)graph on \( v \) points with \( \lambda \) multiple edges for each pair of points and \( j \) loops at each vertex, into ordered blocks \( (a_1, a_2, \ldots, a_{k-1}, a_1) \). Each block is the subgraph which contains both the set of unordered edges \( \{a_i, a_j\} \), for each pair of consecutive edges in the ordered list, and also the loop at vertex \( a_1 \).

We investigate the existence of fixed point families for the eccentric digraph (\( \text{ED} \)) operator, which was introduced in \([1]\). In \([2]\), the notion of the period \( \rho(G) \) of a digraph \( G \) (under the \( \text{ED} \) operator) was defined, and it was observed, but not proved, that for any odd positive integer \( m \), \( C_m \times C_m \) is periodic, and that \( \rho(\text{ED}(C_m \times C_m)) = 2\rho(\text{ED}(C_m)) \). Also in \([2]\), the following question was posed: which digraphs are fixed points under the digraph operator? We provide a proof for the observations about \( C_m \times C_m \), and in the process show that these products comprise a family of fixed points under \( \text{ED} \). We then provide a number of other interesting examples of fixed point families.

In this paper, we derive some necessary existence conditions for balanced arrays (B-arrays) of strength eight and with two levels by making use of some classical inequalities such as Cauchy, Hölder, and Minkowski. We discuss the usefulness of these conditions in the study of the B-arrays, and also present some illustrative examples.

For a graph \( G \) having chromatic number \( k \), an equivalence relation is defined on the set \( X \) consisting of all proper vertex \( k \)-colorings of \( G \). This leads naturally to an equivalence relation on the set \( \mathcal{P} \) consisting of all partitions of \( V(G) \) into \( k \) independent subsets of color classes. The notion of a partition type arises and the algebra of types is investigated.

We derive a new upper bound of \( 26 \) for the Ramsey number \( R(K_5 – P_3, K_5) \), lowering the previous upper bound of \( 28 \). This leaves \( 25 \leq R(K_5 – P_5, K_5) \leq 26 \), improving on one of the three remaining open cases in Hendry’s table, which listed Ramsey numbers for pairs of graphs \( (G, H) \) with \( G \) and \( H \) having five vertices.

We also show, with the help of a computer, that \( R(B_2, B_6) = 17 \) and \( R(B_2, B_7) = 18 \) by full enumeration of \( (B_2, B_6) \)-\({good}\) graphs and \( (B_2, B_7) \)-\({good}\) graphs, where \( B_n \) is the book graph with \( n \) triangular pages.

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 \( v_f(i) = \lvert \{v \in V(G) : f^+(v) = i\} \rvert \) and let \( e_f(i) = \lvert \{e \in E(G) : f(e) = i\} \rvert \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( \lvert e_f(0) – e_f(1) \rvert \leq 1 \). 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 \( L \)-products of cycles with stars.

In [A.G. Chetwynd and A.J.W. Hilton, Critical star multigraphs, Graphs and Combinatorics 2(1986), 209-221], Chetwynd and Hilton started the investigations of the edge-chromatic properties of a particular class of multigraphs, which they called star multigraphs. A star multigraph is a multigraph such that there exists a vertex \( v^* \) that is incident with each multiple edge. Star multigraphs turn out to be useful tools in the study of the chromatic index of simple graphs.

The main goal of this paper is to provide shorter and simpler proofs of all the main theorems contained in the above-mentioned paper. Most simplifications are achieved by means of a formula for the chromatic index recently obtained by the author and by a careful use of arguments involving fans.

The existence of an equivalence subset of rational functions with Fibonacci numbers as coefficients and the Golden Ratio as fixed point is proven. The proof is based on two theorems establishing basic relationships underlying the Fibonacci Sequence, Pascal’s Triangle, and the Golden Ratio.

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.