A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size 2 or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi’_F(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). It has been shown that these concepts generalize both edge domination and matchings in graphs. In this paper, we consider the two color frames \( Y_1 \) and \( Y_2 \) that result from the claw \( K_{1,3} \), where \( Y_1 \) has exactly one red edge and \( Y_2 \) has exactly two red edges. An edge \( e \) in a graph \( G \) is a non-claw edge if \( e \) belongs to no claw in \( G \). It is shown that if \( G \) is a connected graph containing \( \ell \) non-claw edges, then \( \chi’_{Y_1}(G) \leq \chi’_{Y_2}(G) \leq 3\chi’_{Y_1}(G) – 2\ell \) and \( \chi’_{Y_1}(G) = \chi’_{Y_2}(G) \) if and only if \( G \) is a path or cycle. Furthermore, a pair \( a, b \) of positive integers can be realized as the \( Y_1 \)-chromatic index and \( Y_2 \)-chromatic index for some connected graph of order at least 4 if and only if \( a \leq b \leq 3a \) and \( b \geq 2 \).

We prove nonexistence of circulant weighing matrices with parameters from seven previously open entries of the updated Strassler’s table. The method of proof utilizes some modular constraints on circulant weighing matrices with multipliers.

For a connected graph \( G = (V, E) \) of order at least two, a chord of a path \( P \) is an edge joining two non-adjacent vertices of \( P \). A path \( P \) is called a monophonic path if it is a chordless path. A longest \( x-y \) monophonic path is called an \( x-y \) detour monophonic path. A set \( S \) of vertices of \( G \) is a monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) monophonic path for some elements \( x \) and \( y \) in \( S \). The minimum cardinality of a monophonic set of \( G \) is the monophonic number of \( G \), denoted by \( m(G) \). A set \( S \) of vertices of \( G \) is a detour monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) detour monophonic path for some \( x \) and \( y \) in \( S \). The minimum cardinality of a detour monophonic set of \( G \) is the detour monophonic number of \( G \) and is denoted by \( dm(G) \). We determine bounds for it and characterize graphs which realize these bounds. Also, for each pair \( a, b \) of integers with \( 2 \leq a \leq b \), we prove that there is a connected graph \( G \) with \( m(G) = a \) and \( dm(G) = b \).

Fault diagnosis, testing and tolerance in large scale computer and communication systems is a topic of great interest to the computer and communications research communities. In this paper, we give a broad survey of an area called system level diagnosis initiated by Preparata, Metze and Chien. Our survey includes different models of diagnosis and related diagnosis and diagnosability algorithms. In particular, we have given a detailed view of distributed diagnosis. We believe most of these works form the foundation of the research in the emerging area of fault tolerance in a mobile environment.

For vertices \( u \) and \( v \) in a connected graph \( G = (V, E) \), the monophonic detour distance \( d_m(u, v) \) is the length of a longest \( u-v \) monophonic path in \( G \). An \( u-v \) monophonic path of length \( d(u, v) \) is an \( u-v \) monophonic detour or an \( u-v \) \( m \)-detour. The set \( I_{d_m}[u, v] \) consists of all those vertices lying on an \( u-v \) \( m \)-detour in \( G \). Given a set \( S \) of vertices of \( G \), the union of all sets \( I_{d_m}[u, v] \) for \( u, v \in S \), is denoted by \( I_{d_m}[S] \). A set \( S \) is an \( m \)-detour convex set if \( I_{d_m}[S] = S \). The \( m \)-detour convex hull \( [S]_{d_m} \) of \( S \) in \( G \) is the smallest \( m \)-detour convex set containing \( S \).

A set \( S \) of vertices of \( G \) is an \( m \)-detour set if \( I_{d_m}[S] = V \) and the minimum cardinality of an \( m \)-detour set is the \( m \)-detour number \( md(G) \) of \( G \). A set \( S \) of vertices of \( G \) is an \( m \)-detour hull set if \( [S]_{d_m} = V \) and the minimum cardinality of an \( m \)-detour hull set is the \( m \)-detour hull number \( md_h(G) \) of \( G \).

Certain general properties of these concepts are studied. Bounds for the \( m \)-detour hull number of a graph are obtained. It is proved that every two integers \( a \) and \( b \) with \( 2 \leq a \leq b \) are realizable as the \( m \)-detour hull number and the \( m \)-detour number respectively, of some graph. Graphs \( G \) of order \( n \) for which \( md_h(G) = n \) or \( md_h(G) = n-1 \) are characterized. It is proved that for each triple \( a \), \( b \), and \( k \) of positive integers with \( a < b \) and \( k \geq 3 \), there exists a connected graph \( G \) with \( rad_m(G) = a \), \( diam_m(G) = b \), and \( md_h(G) = k \).

For a connected graph \( G \) of order \( n \geq 2 \), a set \( S \) of vertices of \( G \) is a geodetic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x \)-\( y \) geodesic for some elements \( x \) and \( y \) in \( S \). The geodetic number \( g(G) \) of \( G \) is the minimum cardinality of a geodetic set of \( G \). A geodetic set of cardinality \( g(G) \) is called a \( g \)-set of \( G \).

A set \( S \) of vertices of a connected graph \( G \) is an open geodetic set of \( G \) if for each vertex \( v \) in \( G \), either \( v \) is an extreme vertex of \( G \) and \( v \in S \); or \( v \) is an internal vertex of an \( x \)-\( y \) geodesic for some \( x, y \in S \). An open geodetic set of minimum cardinality is a minimum open geodetic set, and this cardinality is the open geodetic number, \( og(G) \).

A connected open geodetic set of \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) is connected. The minimum cardinality of a connected open geodetic set of \( G \) is the connected open geodetic number of \( G \) and is denoted by \( og_c(G) \).

A total open geodetic set of a graph \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) contains no isolated vertices. The minimum cardinality of a total open geodetic set of \( G \) is the total open geodetic number of \( G \) and is denoted by \( og_t(G) \). A total open geodetic set of cardinality \( og_t(G) \) is called an \( og_t \)-set of \( G \).

Certain general properties satisfied by total open geodetic sets are discussed. Graphs with total open geodetic number \( 2 \) are characterized. The total open geodetic numbers of certain standard graphs are determined. It is proved that for positive integers \( r \), \( d \), and \( k \geq 4 \) with \( r \leq d \leq 2r \), there exists a connected graph of radius \( r \), diameter \( d \), and total open geodetic number \( k \). It is also proved that for the positive integers \( a \), \( b \), and \( n \) with \( 4 \leq a \leq b \leq n \), there exists a connected graph \( G \) of order \( n \) such that \( og_t(G) = a \) and \( og_c(G) = b \).

In this paper, we provide a powerful technique for the existence of Hamilton-Waterloo Problem from lower order to higher order.

In 1996, Muthusamy and Paulraja conjectured that for \( k \geq 3 \), the Cartesian product \( K_m \Box K_n \) has a \( P_k \)-factorization if and only if \( mn \equiv 0 \mod k \) and \( 2(k-1) | k(m+n-2) \). Recently, Chitra and Muthusamy have partially settled this conjecture for \( k = 3 \). In this paper, it is shown that for \(k = 4\) the above conjecture is true if \((m\mod 12, n \mod 12)\) \in \(\{(0,2), (2,0), (0,8), (8,0), (2,6), (6,2), (6,8), (8,6), (4,4)\}\). The left over cases for \(k = 4\) are \((m\mod 12, n\mod 12) \)\in \(\{(0,5), (5,0), (0,11), (11,0), (1,4), (4,1), (3,8), (8,3), (4,7), (7,4), (4,10), (10,4), (8,9), (9,8), (10,10)\}\).

In the framework of P systems introduced by Paun (1998), the generation of rectangular arrays and hexagonal arrays has been studied in the literature. In this paper, we introduce a new P system generating a family of hexagonal array languages. We compare this new family with the existing families of hexagonal array languages.

Hypertournaments are generalizations of tournaments. We discuss the concept of scores, losing scores, total scores, and degrees in \(k\)-hypertournaments and present characterizations of sequences to be score, losing score, total score, and degree sequences of some \(k\)-hypertournaments. We further discuss stronger upper and lower bounds for scores and losing scores. We extend the concept of scores, losing scores, and degrees to bipartite hypertournaments. In the end, we list some open problems in hypertournaments.