Structural Characteristics of Two-Sender Index Coding

This paper studies index coding with two senders. In this setup, source messages are distributed among the senders possibly with common messages. In addition, there are multiple receivers, with each receiver having some messages a priori, known as side-information, and requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate normalized codelength, which is expressed as the optimal broadcast rate. In this work, firstly, for a given TSUIC problem, we form three independent sub-problems each consisting of the only subset of the messages, based on whether the messages are available only in one of the senders or in both senders. Then, we express the optimal broadcast rate of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we discover the structural characteristics of TSUIC. For the proofs of our results, we utilize confusion graphs and coding techniques used in single-sender index coding. To adapt the confusion graph technique in TSUIC, we introduce a new graph-coloring approach that is different from the normal graph coloring, which we call two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. We further determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates. Finally, we generalize the results of a class of TSUIC problems to multiple senders.

On normal graph of a finite group

Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A U B ? N. The aim of this paper is to study the normal graph of a finite group G. It is proved, among other things, that the groups with identical character table have isomorphic normal graphs and so this new graph associated to a group has good relationship by its group structure. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.

Vertex and edge connectivity of non-normal graphs

Let [Formula: see text] be a finite group and [Formula: see text] be a subgroup of [Formula: see text]. The non-normal graph of [Formula: see text] in [Formula: see text], denoted by [Formula: see text], is defined as the bipartite graph with two parts [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are the normalizer and the core of [Formula: see text] in [Formula: see text], respectively. Two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In this paper, we consider vertex and edge connectivity of [Formula: see text]. We show that [Formula: see text] and if [Formula: see text] is a positive integer such that [Formula: see text] and [Formula: see text], then the graph [Formula: see text] has a cycle of length [Formula: see text].

NON-NORMAL GRAPHS OF FINITE GROUPS

Let G be a finite group and H be a subgroup of G. We introduce the non-normal graph of H in G, denoted by [Formula: see text]H,G, and give some of the graph theoretical properties of [Formula: see text]H,G.