CHAPTER 1 Introduction 1.1 Graphs In this thesis by a graph we mean a simple graph G = (V; E), where V is a finite nonempty set and E is a subset of the set of all 2-subsets of V . The elements in V and E are called the vertices and edges respectively. Sometimes the vertex and the edge sets of G are denoted by V (G) and E(G) respectively. The total number of vertices in V (G) is called the order of G. Two vertices x and y are said to be adjacent if {x; y} is an edge; otherwise they are non-adjacent. If e = {x; y} is an edge then both x and y are called the end vertices of e. If two vertices are Copyrightadjacent then each is called a neighbour of the other vertex. The complement of a graph G is the graph G on the same vertices of G such that two vertices of G are adjacent if and only if they are not adjacent in G. A graphIITG is called a completeKharagpur graph if for any two vertices x and y in G, {x; y} is an edge. In this thesis we denote a complete graph with n vertices by the symbol Kn. A graph G is called a bipartite graph if the vertex set V (G) can be Introduction partitioned into two nonempty subsets X and Y in such a way that each edge of G has one end vertex in X and another end vertex in Y . The partition {X; Y } is called a bipartition of G.A complete bipartite graph G is a bipartite graph with bipartition V (G) = X ∪ Y , such that every vertex in X is adjacent to every vertex in Y . If ∣X∣ = m and ∣Y ∣ = n, then a complete bipartite graph is denoted by Km;n. A graph G is called t-partite graph if the vertex set V (G) can be partitioned into two nonempty subsets X1;X2;:::;Xt such that each edge of G has one end vertex in Xi and the other end vertex in Xj, 1 ≤ i ≠ j ≤ t.A complete t-partite graph is a t-partite graph with partition V (G) = X1 ∪ X2 ∪ ⋅ ⋅ ⋅ ∪ Xt such that every vertex in Xi is adjacent to every vertex in Xj, 1 ≤ i ≠ j ≤ t. If ∣Xi∣ = ni, for i = 1; 2; : : : ; t, then a complete t-partite graph is denoted by Kn1;n2;:::;nt . A graph H = (V1;E1) is said to be a subgraph of a graph G = (V; E) if V1 ⊆ V and E1 ⊆ E. Further H is called an induced subgraph if H contains all the edges of G whose end vertices are in H. The degree of a vertex x in a graph G is the number of vertices in G which are adjacent to x. In this thesis we denote the degree of a vertex x in a graph G by degG(x) (or simply deg(x), if the graph is understood). A degree one vertex is called a pendant vertex or a leaf. A walk in a graph G is a finite sequence W ∶ v0e1v1e2v2 : : : vk−1ekvk whose terms are alternately vertices and edges (starting and ending with vertices) such that, for 1 ≤ i ≤ k, the edge ei has end vertices vi−1 and vi. If v0 = vk then W is called a closed walk. If the vertices v0; v1; : : : ; vk of the walk W ∶ v0e1v1e2v2 : : : vk−1ekvk − Copyrightare distinct then W is called a path (or a v0 vk path). A path on n vertices is denoted by Pn. The number of edges in a path is called its length. If G is a connected graph then the distance between two vertices u and v in G, denoted ( ) ( ) by dG u; v IIT(or simply d u; v Kharagpurif the graph is understood), is the minimum of the lengths of all u − v paths in G. A graph G is said to be connected if each pair of vertices in G belongs to a path; otherwise G is called disconnected. The 2 1.1 Graphs eccentricity of a vertex v in a connected graph G is denoted by e(v) and defined as e(v) = max{d(u; v) ∶ u ∈ V (G)}. The diameter of a connected graph G, denoted by diam(G), is given by max{e(v) ∶ v ∈ V (G)}.A cycle is a closed walk W ∶ v0e1v1e2v2 : : : vk−1ekv0 where all the vertices are distinct. A cycle with n vertices is denoted by Cn.A tree is a connected graph without any cycle. A star is a tree having a vertex which is adjacent to all the other vertices. A star with n vertices is the bipartite graph K1;n−1.A rooted tree is a tree with one vertex chosen as the root. A complete m-ary tree is a rooted tree such that the degree of the root vertex is m, the degree of all other non-pendant vertices is equal to m + 1, and all pendant vertices are of the same distance from the root. A tree T is said to be a m-distant tree if there is a path P of maximum length in T such that every vertex in T is of distance at the most m from P . This path P is called a central path of T . Every tree is an m-distant tree for some m. An 1-distant tree is called a caterpillar and a 2-distant tree is called a lobster. The n-dimensional cube or hypercube Qn is the graph whose vertices are the n-tuples with entries in {0; 1} and its edges are the pairs of n-tuples that differ in exactly one position. The cartesian product G12G2 of graphs G1 and G2 is the graph whose vertex set is the cartesian product V (G1) × V (G2), and any two vertices (u1; v1) and (u2; v2) are adjacent if and only if either u1 = u2 and v1 is adjacent to v2 in G2 or v1 = v2 and u1 is adjacent to u2 in G1. One can easily verify that G12G2 = G22G1. The hypercube Qn can be expressed as the cartesian product of n copies of K2. The book graph Bm is defined as the cartesian product Sm+12P2, Copyrightwhere S + is the star graph K . The cartesian product S + 2P is called the m 1 1;m m 1 n stacked book graph Bm;n. The prism graph Ym is the cartesian product Cm2P2. A two dimensional grid graph is the cartesian product Pm2Pn. For any graph G and a positiveIIT integer r, the rthKharagpurpower of G, denoted by Gr, is the graph with the vertex set same as that of G and two vertices u and v are adjacent in Gr if and 3 Introduction ( ) ≤ { ⌊ n ⌋} only if dG u; v r. For any list l chosen from 1; 2; 3;:::; 2 , a circulant graph Cin(l) is a graph on the vertices v0; v1; : : : ; vn−1 such that each vi, 0 ≤ i ≤ n − 1, is adjacent to vi+j and vi−j (subscripts are taken mod n) for every j in the list l. ( ⌊ n ⌋) The circulant graph Cin 1; 2;:::; 2 is the complete graph Kn and the graph Cin(1) is the cycle Cn. The generalized Petersen graph GP (n; r), for n ≥ 3 and ≤ ≤ ⌊ n−1 ⌋ { } 1 r 2 , is the graph with vertex set u0; u1; : : : ; un−1; v0; v1; : : : ; vn−1 and edge set {{ui; ui+1}; {ui; vi}; {vi; vi+r}; i = 0; 1; 2; : : : ; n − 1}, where subscripts are to be read modulo n. 1.2 Motivation for Radio k-colorings of Graphs The Frequency Assignment Problem (FAP) is the problem of assigning frequencies to transmitters in some optimal manner that avoids interferences. Nearly three decade back this problem has been modeled as a graph coloring (labeling) problem see, Hale (1980). This coloring has several variations depending upon the type of assignment of frequency to transmitters. In this section, we discuss some of them including radio k-coloring of graphs. The FAP plays an important role in wireless network and is a well-studied interesting problem. Due to rapid growth of wireless networks and to the relatively scarce radio spectrum the importance of FAP is growing significantly. Many researchers have modeled FAP as an optimization problem as follows: Given Copyrighta collection of transmitters to be assigned operating frequencies and a set of interference constraints on transmitter pairs, find an assignment that satisfies all the interference constraints and minimizes the value of a given objective function. HaleIIT(1980) has modeledKharagpur FAP as both frequency-distance constrained and frequency constrained optimization problems. Since then, a number of graph colorings have been inspired by the FAP. 4 1.2 Motivation for Radio k-colorings of Graphs T -colorings of graphs is also a model of FAP, studied by Liu (1992, 1994, 1996, 2000), Cozzens et al. (1982, 1991), Griggs et al. (1994), Rabinowitz et al. (1985), Raychaudhuri (1994) and Tesman (1989). For any simple connected graph G and a list T of non-negative integers containing 0, T -coloring is a mapping f from the vertex set V of G to the set of non-negative integers {0; 1; 2;::: } such that ∣f(x) − f(y)∣ ∉ T whenever x and y are adjacent in G. The span of a T -coloring f is the difference between the largest and the smallest numbers in f(V ), i.e., max{∣f(x) − f(y)∣ ∶ x; y ∈ V }.