We show that the butterfly network and Benes network can be embedded into generalized fat trees with minimum dilation.

The crossing number of a graph \( G \) is the minimum number of crossings of its edges among the drawings of \( G \) in the plane and is denoted by \( \operatorname{cr}(G) \). In this paper, we obtain bounds for the crossing number for two different honeycomb tori, namely, the honeycomb rectangular torus and the honeycomb rhombic torus, which are obtained by adding wraparound edges to honeycomb meshes.

In cellular radio communication systems, the concept of maximum packing is used for dynamic channel assignment. An \( H \)-packing of a graph \( G \) is a set of vertex-disjoint subgraphs of \( G \), each of which is isomorphic to a fixed graph \( H \). The maximum \( H \)-packing problem is to find the maximum number of vertex-disjoint copies of \( H \) in \( G \), called the packing number, denoted by \( \lambda(G, H) \). In this paper, we determine the maximum \( H \)-packing number of hexagonal networks when \( H \) is isomorphic to \( P_6 \) as well as \( K_{1,3} \).

A kernel in a directed graph \( D(V, E) \) is a set \( S \) of vertices of \( D \) such that no two vertices in \( S \) are adjacent and for every vertex \( u \) in \( V \setminus S \), there is a vertex \( v \) in \( S \) such that \( (\overline{u, v}) \) is an arc of \( D \). The problem of existence of a kernel is NP-complete for a general digraph. In this paper, we introduce the acyclic kernel problem for an undirected graph \( G \) and solve it in polynomial time for certain cycle-related graphs.

A kernel in a directed graph \( D(V, E) \) is a set \( S \) of vertices of \( D \) such that no two vertices in \( S \) are adjacent and for every vertex \( u \) in \( V \setminus S \), there is a vertex \( v \) in \( S \) such that \( (u, v) \) is an arc of \( D \). The problem of existence of a kernel is NP-complete for a general digraph. In this paper, we solve the strong kernel problem of an oriented biregular graph in polynomial time.

String-token Petri net, which is a variation of coloured Petri net, has been introduced in [1] by requiring the tokens to be labeled by strings. Languages in regular and linear families, which are two basic classes in the Chomsky hierarchy, are generated by these Petri nets [2]. An extension called array-token Petri net, introduced in [5] by labeling tokens by arrays, generates picture languages. Properties related to generative power of array-token Petri net are considered in [3]. In this paper, application of array-token Petri net to generate English alphabetic letters treated as rectangular arrays is examined.

Given a graph \( G = (V, E) \), a set \( W \subseteq V \) is said to be a resolving set if for each pair of distinct vertices \( u, v \in V \), there is a vertex \( x \) in \( W \) such that \( d(u, x) \neq d(v, x) \). The resolving number of \( G \) is the minimum cardinality of all resolving sets. In this paper, a condition is imposed on resolving sets and a conditional resolving parameter is studied for grid-based networks.

Let \( G = (V, E) \) be a graph. A vertex labeling \( f: V \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E \). For each \( i \in \mathbb{Z}_2 \), define \( v_f(i) = |f^{-1}(i)| \) and \( e_f(i) = |{f^*}^{-1}(i)| \). We call \( f \) friendly if \( |v_f(1) – v_f(0)| \leq 1 \). The full friendly index set of \( G \) is the set of all possible values of \( e_f(1) – e_f(0) \), where \( f \) is a friendly labeling. In this paper, we study the full friendly index set of the wheel \( W_n \), the tensor product of paths \( P_2 \) and \( P_n \), i.e., \( P_2 \otimes P_n \), and the double star \( D(m, n) \).

The detour order of a graph \( G \), denoted \( \tau(G) \), is the order of a longest path in \( G \). A partition \( (A, B) \) of \( V(G) \) such that \( \tau(\langle A \rangle) \leq a \) and \( \tau(\langle B \rangle) \leq b \) is called an \( (a, b) \)-partition of \( G \). A graph \( G \) is called \( \tau \)-\textit{partitionable} if \( G \) has an \( (a, b) \)-partition for every pair \( (a, b) \) of positive integers such that \( a + b = \tau(G) \).

The well-known Path Partition Conjecture states that every graph is \( \tau \)-partitionable. Motivated by the recent result of Dunbar and Frick [6] that if every \( 2 \)-connected graph is \( \tau \)-partitionable, then every graph is \( \tau \)-partitionable, we show that the Path Partition Conjecture is true for a large family of \( 2 \)-connected graphs with certain ear-decompositions. Also, we show that a family of \( 2 \)-edge-connected graphs with certain ear-decompositions is \( \tau \)-partitionable.

he problem of determining the collaboration graph of co-authors of Paul Erdos is a challenging task. Here we take up this problem for the case of Rolf Nevanlinna Prize Winners. Even though the number of prize winners as of date is 7, the collaboration graph has 20 vertices and 41 edges and possesses several interesting properties. In this paper, we have obtained this graph and determined standard graph parameters for the graph as well as its complement besides probing its structural properties. Several new results were obtained.