Chapter 1
Introduction
1.1 Background
In 1735, Swiss mathematician Leonhard Euler solved the “Ko¨nigsberg bridge problem”. This gave birth to Graph Theory and Leonhard Euler is regarded as the father of Graph Theory. The paper titled “Solutio problematis ad geometriam situs pertinentis” written by L. Euler [6] on the Seven Bridges of Ko¨nigsberg and published in 1741 is regarded as the first paper in the history of Graph theory.
A graph consists of objects and the relationship between the objects. A graph comprises of vertices (also called nodes or points) representing the objects and edges (also called arcs or lines) joining two vertices if they are related. A graph is denoted by G = (V, E) where V (G) is the vertex set and E(G) is the edge set. The order of a graph is the cardinality of its vertex set and size of a graph is the cardinality of its edge set. A loop is an edge having same end vertices [16]. Multiple edges are the edges which share same end points [16]. A graph without loops and multiple edges is called a “simple graph”[16]. Otherwise the graph is called a “pseudo graph”. For example, see Figure 1.1.
Figure 1.1: Simple and Pseudo graph
In the past few decades, Graph theory has been extensively applied in the field of Computer science, Chemistry, Sociology, Electrical engineering, Communication science etc. One of the most important graph theoretical concepts which has vast appication in day to day life is graph coloring. Some of the important applications of graph coloring could be seen in circuit design, astronomy, coding theory, data base management, communication network, channel assignment, etc.
Graph coloring is a function from the vertex set V (G) to the set of colors such that no two adjacent vertices receive the same color. In other words, it is a way of coloring the vertices of a graph such that any two adjacent vertices receive different colors. This is called a “proper vertex coloring”. While trying to color a map of the counties of England, Francis Guthrie postulated the “Four color conjecture” which stated that four colors were sufficient to color the map so that no regions sharing a common border receive the same color. The Four color conjceture was proved in 1976 by Kenneth Appel and Wolfgang Haken.
Over the course of time, different types of coloring have been developed. Some of them being T-coloring, L(2,1)-coloring, Radio coloring, Hamiltonian coloring, Total coloring etc. An L(2,1) coloring is an assignment of colors such that the color difference between the adjacent vertices is atleast two and the color difference between the vertices at distance two is atleast one.
1.2 Preliminaries
Every graph referred to in this project work is finite, undirected, simple, connected graph.
Definition 1.2.1. A graph G is an ordered pair (V, E) consisting of a set V of vertices and a set E of edges which are 2-element subsets of V . If the edge is an unordered pair then the graph is undirected.
Definition 1.2.2. A path of length k in a graph is an alternating sequence of distinct vertices and edges, v0, e0, v1, e1, v2, …, vk−1, ek−1, vk. We denote a path on n vertices by Pn.
Example:
Figure 1.2: A path on 5 vertices (P5)
Definition 1.2.3. A cycle graph is a path with the first and last vertices coinciding. We denote a cycle graph on n vertices by Cn.
Example:
Figure 1.3: A cycle on 5 vertices (C5)
Definition 1.2.4. A wheel graph on n vertices is a join of two graphs namely, complete graph
K1 and cycle Cn−1. We denote a wheel graph on n vertices by Wn. So, Wn = K1 + Cn−1
Example:
Figure 1.4: A wheel on 6 vertices (W6)
Definition 1.2.5. A fan graph on n vertices is a join of two graphs namely, complete graph K1
and a path Pn−1. We denote a fan graph on n vertices by Fn. So, Fn = K1 + Pn−1. Example:
Figure 1.5: A fan on 6 vertices (F6)
Definition 1.2.6. A star graph is a tree formed by joining a single vertex to pendant vertices(i.e. vertices of degree one). We denote a star graph on n vertices by K1,n−1.
Example:
Figure 1.6: A star on 5 vertices (K1,4)
Definition 1.2.7. A bistar Br,s on r + s + 2 vertices is a tree obtained by joining the central vertices of two stars K1,r and K1,s.
Example:
Figure 1.7: A bistar on 10 vertices (B3,5)
Definition 1.2.8. A prism graph GP (n, 1) is a planar graph obtained by the cartesian product of a cycle Cn and K2. So, GP (n, 1) = Cn K2.
Example:
Figure 1.8: A prsim graph on 16 vertices (GP (8, 1))
Definition 1.2.9. [16] Given a simple graph G = (V, E), the complement of G denoted by G is a simple graph with the vertex set V (G) and any two vertices are adjacent in G if and only if they are not adjacent in G.
Definition 1.2.10. [4] Given a simple connected graph G = (V, E), an L(2, 1)-coloring is a function f from the vertex set V (G) to the set of non-negative integers such that:
(i) |f(u) − f(v)| ≥ 2 ∀ uv ∈ E(G)
(ii) |f(u) − f(v)| ≥ 1 if d(u, v) = 2
Example:
Figure 1.9: An L(2, 1)-coloring of P5
Definition 1.2.11. [12] Let f be an L(2, 1)-coloring of a graph G. The span of f, denoted by
λ(f) is the maximum color assigned to a vertex in G by f.
Definition 1.2.12. [4] The span of G, denoted by λ(G) is the smallest among all the f-spans.
λ(G) = min{λ(f) : f is an L(2,1)-coloring}
Definition 1.2.13. [12] A span coloring is an L(2, 1)-coloring in which the maximum color used is λ(G).
Figure 1.10: Span colorings of P5
Definition 1.2.14. [12] A span coloring f is said to be a full-coloring if f is onto.
Definition 1.2.15. [12] An L(2, 1)-coloring f is irreducible if there does not exist an L(2, 1)- coloring g such that g(u) ≤ f(u) for all u ∈ V (G) and g(v) < f(v) for some v ∈ V (G). Example:
Here, the first span coloring is reducible since the color 4 given to the right most vertex can be reduced to 2. The second span coloring is an irreduclible span coloring of P5.
Figure 1.11: Reducible and irreducible span colorings of P5
Definition 1.2.16. [12] Let f be an L(2, 1)-coloring of a graph G using the colors {0, …, k}. An integer h such that 0 < h < k is said to be a hole in f if f assigns the color h to none of the vertices in G. Let H denote the set of holes. Let Hλ(G) denote the maximum number of holes in an irreducible span coloring of G.
Example:
Figure 1.12: Irreducible span colorings of P5 with different set of holes
Here, the first irreducible span coloring has two holes namely, 1 and 3. The second irre- ducible span coloring has one hole namely, 2. The third irreducible span coloring has no holes. So, the maximum number of holes is 2.
Definition 1.2.17. [12] The index of G is the minimum number of unused colors in a span coloring of G. It is denoted by ρ(G).
Definition 1.2.18. [12] An L(2, 1)-coloring using colors from 0 to k is a no-hole coloring if it uses all the labels from 0 to k.
Definition 1.2.19. [12] An inh-coloring is defined as an irreducible no-hole coloring.
Definition 1.2.20. [12] Let f be an irreducible span coloring of a graph G. If f has the maximum number of holes among all the irreducible span colorings of G, then f is called a maximum-hole coloring.
1.3 Literature Review
The channel assignment problem was propsed by W.K. Hale [10] in 1980. Inspired by the chan- nel assignment problem, F. Roberts as a of his communication to J. Griggs in 1988, proposed the problem of optimally assigning frequencies to radio transmitters using nonnegative integers to represent channels, so that “close” stations are assigned different integers, and difference in the frequencies assigned to “very close” staions is atleast 2. This gave rise to the study of L(2, 1)-coloring of a graph which was first proposed by J. Griggs and R. Yeh [9] in 1992.
The concept of no hole 2 distance-coloring was introduced by F. Roberts [14] in 1993 for T -coloring of graphs. Here, the vertices of a graph are colored with positive integers so that the color difference between any two adjacent vertices is greater than 1 and so that the set of integers used as colors is a consecutive set (the no-hole requirement). F. Roberts [14] has studied what graphs have such colorings. He also studied what graphs have such colorings which are near-optimal, that is, separation between the largest and smallest colors used is no more than one larger than the minimum span in a non-adjacent coloring which may have holes.
D. Sakai and C. Wang [15] in 1993 studied a generalization of the no-hole 2-distant color- ings introduced by F. Roberts [14]. Here, the vertices of a graph are colored with consecutive positive integers so that the color difference between any two adjacent vertices get is greater than a fixed positive integer r > 1.
A.A. Bertossi and M.A. Bouccelli [2] have studied how to avoid hidden collision in the multi hop radio networks in 1995.
In the year 2000, P. Fishburn and F. Roberts [7] proved that all trees except stars are full- colorable. They have proved that C6 is the only connected graph with minimum number of vertices exceeding λ(G) which is not full-colorable. They have given a number of other con- nected graphs that are not full-colorable and have given details on full-colorability of graphs with maximum degree four or less than four.
P. Fishburn and F. Roberts [8] in 2003 introduced the parameter µ(G), which is defined to be the minimum integer k for which G has a no-hole coloring, if it exists. They have proved two main results in their paper. First, for every m ≥ 1 there is a graph G with ρ(G) = m and µ(G) = λ(G) + m. Second, for every m ≥ 2 there is a connected G with λ(G) = 2m, n = λ(G) + 2 and ρ(G) = m.
In the year 2007, S.S. Adam et al. [1] have shown that for each k ≥ 2, there exists an
r-regular graph G with ρ(G) = k so that ρ(G) and r are relatively prime integers.
R. Laskar et al. [13] in 2009 have shown that for any tree T that is not a star, inh-span of T is equal to its span. Here, the inh-span of a graph G, denoted by λinh(G), is the smallest number k such that there is an irreducible no-hole L(2, 1)-coloring of G using the integers 0, …, k.
In 2009, R. Laskar and G. Eyabi [12] have determined the exact values of maximum number of holes in the span colorings of certain classes of graphs such as paths, cycles, trees, complete bipartite graphs, complete multi-partite graphs. Also, they have characterized complete graphs in terms of their maximum number of holes.
1.4 Scope of Work
We review the paper “Holes in L(2, 1)-Coloring of Some Classes of Graphs” by R. Laskar and G. Eyabi [12]. We further extend their work by determining the exact values of maximum number of holes in the span colorings of complement of a cycle, wheel, fan, bistar. We also determine the bounds for maximum number of holes in span coloring for prism graph in this project. This work can be further extended by finding the exact values for maximum number of holes for prism graph and some other complex classes of graphs such as generalized Petersen graph, flower graph, different types of product of graphs etc.
Chapter 2
Holes in L(2,1)-Coloring of graphs
Now a days, the assignment of radio frequency to the radio stations is creating a lot of problems because day by day, the number of radio stations getting installed are increasing . As a result, there is an increase in the request for radio frequencies . Interference during communication is a difficulty in assigning channels to different radio stations. Two adjacent stations which have same frequency will have a direct interference or “direct collision” whereas station nearby the adjacent station perform hidden interference or “hidden collision”. Bertossi et al. [3, 2] studied the case of avoiding hidden collision in the multi hop radio networks. To avoid collisions from its neighbor and next to neighbour station we require distinct labeling.
The “channel assignment problem” is the task of optimally assigning frequencies to radio transmitters located at various places so that there is interference-free communication. The frequencies assigned to neighboring transmitters have to be significantly different and pairs of transmitters which are slightly farther away are given different frequencies so that there is no interference. Once, it was said that the number of usable frequencies was very much greater than the number of transmitters. However, this is no longer true.
Inspired by the channel assignment problem introduced by W.K. Hale [10] in 1980, F. Roberts introduced the L(2, 1)-coloring problem as a result of his communication to J. Griggs in 1988. As a result of this, J. Griggs and R. Yeh [9] in 1992 introduced the concept of L(2, 1)-
coloring. In this type of coloring, the colors (nonnegative integers) assigned to the vertices of a graph depend not only on whether two vertices are adjacent but also on whether two vertices are at distance 2.
The L(2, 1)-coloring problem is the problem of assigning frequencies (non-negative inte- gers) to radio transmitters so that “close” transmitters receive different frequencies and “very close” transmitters receive frequencies that are at least two frequencies apart. This problem can be modeled using a graph G. The vertices of the graph represent the transmitters. Based on the proximity of the transmitters and the power of the transmissions, edges are placed between vertices to represent possible interference. Two vertices are “very close” if they are adjacent in the graph and “close” if they are at distance two in the graph. The frequencies that are assigned to the radio stations are denoted by the nonnegative integers 0, …, λ.