We define a new fairness notion on edge-colorings, requiring that the number of vertices in the subgraphs induced by the edges of each color are within one of each other. Given a (not necessarily proper) \( k \)-edge-coloring of a graph \( G \), for each color \( i \in \mathbb{Z}_k \), let \( G[i] \) denote the (not necessarily spanning) subgraph of \( G \) induced by the edges colored \( i \). Let \( v_{i}(G) = |V(G[i])| \). Formally, a \( k \)-edge-coloring of a graph \( G \) is said to be vertex-equalized if for each pair of colors \( i, j \in \mathbb{Z}_k \), \( |v_{i}(G) – v_{j}(G)| \leq 1 \). In this paper, a characterization is found for connected graphs that have vertex-equalized \( k \)-edge-colorings for each \( k \in \{2, 3\} \) (see Corollary 4.1 and Corollary 4.2).

Let \( G = (V, E) \) be a graph. The open neighborhood of a vertex \( v \in V \) is the set \( N(v) = \{u \mid uv \in E\} \) and the closed neighborhood of \( v \) is the set \( N[v] = N(v) \cup \{v\} \). The open neighborhood of a set \( S \) of vertices is the set \( N(S) = \bigcup_{v \in S} N(v) \), while the closed neighborhood of a set \( S \) is the set \( N[S] = \bigcup_{v \in S} N[v] \). A set \( S \subset V \) dominates a set \( T \subset V \) if \( T \subseteq N[S] \), written \( S \rightarrow T \). A set \( S \subset V \) is a dominating set if \( N[S] = V \); and is a minimal dominating set if it is a dominating set, but no proper subset of \( S \) is also a dominating set; and is a \( \gamma \)-set if it is a dominating set of minimum cardinality. In this paper, we consider the family \( \mathcal{D} \) of all dominating sets of a graph \( G \), the family \( \mathcal{MD} \) of all minimal dominating sets of a graph \( G \), and the family \( \Gamma \) of all \( \gamma \)-sets of a graph \( G \). The study of these three families of sets provides new characterizations of the distance-2 domination number, the upper domination number, and the upper irredundance number in graphs.

Irregular-spotty-byte error control codes were devised by the author in [2] and their properties were further studied in [3] and [4]. These codes are suitable for semi-conductor memories where an I/O word is divided into irregular bytes not necessarily of the same length. The \(i\)-spotty-byte errors are defined as \(t_i\) or fewer bit errors in an \(i\)-byte of length \(n_i\), where \(1 \leq t_i \leq n_i\) and \(1 \leq i \leq s\). However, an important and practical situation is when \(i\)-spotty-byte errors caused by the hit of high energetic particles are confined to \(i\)-bytes of the same size only which are aligned together or in words errors occur usually in adjacent RAM chips at a particular time. Keeping this view, in this paper, we propose a new model of \(i\)-spotty-byte errors, viz. uniform \(i\)-spotty-byte errors and present a new class of codes, viz. uniform \(i\)-spotty-byte error control codes which are capable of correcting all uniform \(i\)-spotty-byte errors of \(i\)-spotty measure \( \mu \) (or less). The study made in this paper will be helpful in designing modified semi-conductor memories consisting of irregular RAM chips with those of equal length aligned together.

Let \( \mathcal{M} \) denote the set of matrices over some semiring. An upper ideal of matrices in \( \mathcal{M} \) is a set \( \mathcal{U} \) such that if \( A \in \mathcal{U} \) and \( B \) is any matrix in \( \mathcal{M} \), then \( A + B \in \mathcal{U} \). We investigate linear operators that strongly preserve certain upper ideals (that is, linear operators on \( \mathcal{M} \) with the property that \( X \in \mathcal{U} \) if and only if \( T(X) \in \mathcal{U} \)). We then characterize linear operators that strongly preserve sets of tournament matrices and sets of primitive matrices. Specifically, we show that if \( T \) strongly preserves the set of regular tournaments when \( n \) is odd or nearly regular tournaments when \( n \) is even, then for some permutation matrix \( P \), \( T(X) = P^{t}XP \) for all matrices \( X \) with zero main diagonal, or \( T(X) = P^{t}X^{t}P \) for all matrices \( X \) with zero main diagonal. Similar results are shown for linear operators that strongly preserve the set of primitive matrices whose exponent is \( k \) for some values of \( k \), and for those that strongly preserve the set of nearly reducible primitive matrices.

For a poset \( P = (V(P), \leq_P) \), the strict semibound graph of \( P \) is the graph \( ssb(P) \) on \( V(ssb(P)) = V(P) \) for which vertices \( u \) and \( v \) of \( ssb(P) \) are adjacent if and only if \( u \neq v \) and there exists an element \( x \in V(P) \) distinct from \( u \) and \( v \) such that \( x \leq_P u,v \) or \( u,v \leq_P x \). We prove that a poset \( P \) is connected if

In this paper, we revisit LE graphs, find the minimum \( \lambda \) for decomposition of \( \lambda K_n \) into these graphs, and show that for all viable values of \( \lambda \), the necessary conditions are sufficient for LE-decompositions using cyclic decompositions from base graphs.

We determine the signless Laplacian spectrum for the \( H \)-join of regular graphs \( G_1, \ldots, G_p \). We also find an expression and upper bounds for the signless Laplacian spread of the \( H \)-join of regular graphs \( G_1, \ldots, G_p \).

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path \( \pi = v_1, v_2, \ldots, v_{k+1} \) is a downhill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(v_i) \geq \deg(v_{i+1}) \). Conversely, a path \( \pi = u_1, u_2, \ldots, u_{k+1} \) is an uphill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(u_i) \leq \deg(u_{i+1}) \). The downhill domination number of a graph \( G \) is defined to be the minimum cardinality of a set \( S \) of vertices such that every vertex in \( V \) lies on a downhill path from some vertex in \( S \). The uphill domination number is defined as expected. We explore the properties of these invariants and their relationships with other invariants. We also determine a Vizing-like result for the downhill (respectively, uphill) domination numbers of Cartesian products.

Set-to-Set Broadcasting is an information distribution problem in a connected graph, \( G = (V, E) \), in which a set of vertices \( A \), called originators, distributes messages to a set of vertices \( B \) called receivers, such that by the end of the broadcasting process each receiver has received the messages of all the originators. This is done by placing a series of calls among the communication lines of the graph. Each call takes place between two adjacent vertices, which share all the messages they have. Gossiping is a special case of set-to-set broadcasting, where \( A = B = V \). We use \( F(A, B, G) \) to denote the length of the shortest sequence of calls that completes the set-to-set broadcast from a set \( A \) of originators to a set \( B \) of receivers, within a connected graph \( G \). \( F(A, B, G) \) is also called the cost of an algorithm. We present bounds on \( F(A, B, G) \) for weighted and for non-weighted graphs.

Let \( \gamma_c(G) \) denote the connected domination number of the graph \( G \). A graph \( G \) is said to be connected domination edge critical, or simply \( \gamma_c \)-critical, if \( \gamma_c(G + e) < \gamma_c(G) \) for each edge \( e \in E(\overline{G}) \). We answer a question posed by Zhao and Cao concerning \( \gamma_c \)-critical graphs with maximum diameter.