Eccentric Coloring of a Graph

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). An eccentric coloring of a graph G = (V, E) is a function color: V → N such that (i) for all u, v ∈ V , (color(u) = color(v))⇒ d(u, v) > color(u). (ii) for all v ∈ V , color(v) ≤ e(v). The eccentric chromatic number χe ∈ N for a graph G is the lowest number of colors for which it is possible to eccentrically color G by colors: V → {1, 2, . . . , χe}. In this paper, we have considered eccentric colorability of a graph in relation to other properties. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs.


Introduction
Unless mentioned otherwise for terminology and notation the reader may refer [Buckley and Harary, 1990] and [Chartrand and Lesniak, 1996], new ones will be introduced as and when found necessary.In this paper we consider simple undirected graphs without multiple edges and self loops.The order p is the number of vertices in G and size q is the number of edges in G.
The distance d (u,v) between u and v is the length of a shortest path joining u and v.If there exists no path between u and v then we define d(u, v) = ∞.The eccentricity e(u) of u is the distance to a vertex farthest from u.If d(u, v) = e(u)(v u), we say that v is an eccentric vertex of u.The radius rad(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity A chordal graph is a simple graph in which every cycle of length greater than three has a chord.Equivalently, the graph contains no induced cycle of length four or more.The join of two graphs G 1 and G 2 , defined by Zykov [Zykov, 1949] is denoted G 1 + G 2 and consists of G 1 ∪ G 2 and all edges joining V 1 with V 2 .For p ≥ 4, the wheel W p is defined to be the graph K 1 + C p−1 .A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so that every edge has one end in X and one end in Y; such a partition (X, Y) is called a bipartition of the graph, and X and Y its parts.We denote a bipartite graph G with bipartition (X, Y) by G[X, Y].If G[X, Y] is simple and every vertex in X is joined to every vertex in Y, then G is called a complete bipartite graph, denoted by K m,n .Clique in a graph is a set of pairwise adjacent vertices.The clique number of a graph G, written ω(G), is the maximum size of a set of pairwise adjacent vertices (clique) in G.Many researchers have studied the relations related to clique number [Dirac, 1961], [Fulkerson and Gross, 1965], [Matula, 1972], [Voloshin, 1982], [Voloshin and Gorgos, 1982], etc.
In [Sloper, 2004], Sloper has introduced the concept of eccentric coloring of a graph and studied the eccentric coloring of trees.
Definition 1.2 (Sloper, 2004).The eccentric chromatic number χ e ∈ N for a graph G is the lowest number of colors for which it is possible to eccentrically color G by colors: V → {1, 2, . . ., χ e }.

Bounds on the Radius and Diameter of an Eccentric Colorable Graph
In this section we determine some bounds on diameter, radius, size of center, clique number, etc. depending on the eccentric colorability of the graph.
Proposition 2.1.In any eccentric colorable graph, there exists at most one vertex with color c = diam(G).
Proof.Let G be an eccentric colorable graph.Suppose on the contrary, there exist at least two vertices, say, u, v, with color c = diam(G).But d(u, v) ≤ diam(G), which contradicts the definition of eccentric colorable graph.Hence, G contains at most one vertex with color c = diam(G).
In [Negami and XU, 1986], S. Negami et al. have proved that in any 2-self-centered graph, for every vertex v there exists a cycle of length 4 or 5 that preserves the graph distance with a vertex u in that cycle.
Proof.Let G be a graph with diam(G) ≤ 2.Here three cases arise.
In this case the graph is a complete graph, which is not eccentric colorable.
Let u be a central vertex of G. Since e(u) = 1, u can be colored with color 1 only.Let v and w be any two diametral vertices of G.The vertices v and w can not be colored with color 1.Suppose v is colored with color 2, then w can not be colored with color 2, because d(v, w) = 2. Hence, G is not eccentric colorable.Case(iii): rad(G) = 2 and diam(G) = 2.
In this case the graph G is a self centered graph of radius two.Referring to the paper [Negami and XU, 1986] as cited above, every self centered graph of radius two contains at least four vertices and degree of every vertex is at least two.Let u be any vertex of G and v, w are adjacent to u.Since e(u) = e(v) = e(w) = 2, we can use at most two colors, color 1 and color 2 to color these three vertices.Suppose u is colored with color 1 then any of v and w can not be colored with color 1 and both v and w can not be colored with color 2. Hence, G is not eccentric colorable.Now suppose u is colored with color 2 then any of v and w can not be colored with color 2. Hence, v and w are colored with color 1 if v and w are not adjacent(If v and w are adjacent then G is not eccentric colorable).Since deg(v) ≥ 2, there exists a vertex say, x adjacent to v. The vertex x can not be colored with color 1 or color 2, since d(u, x) ≤ 2 and d(v, x) ≤ 1.Hence, G is not eccentric colorable.

Note:
The converse of Theorem 2.1 need not be true.For example, C 7 [K 2 ] is of diameter 3, which is not eccentric colorable.
Remark 1.The following graphs are not eccentric colorable, (i) Connected, self centered, chordal graph, (ii) Petersen graph, (iii) Graph with radius one, Proof.Let a graph G containing the wheel W 1,p−1 be eccentric colorable.The subgraph W 1,p−1 requires at least 2 + ⌈ p−1 2 ⌉ colors as shown in Figure 1.Hence, χ e (G) ≥ 2 + ⌈ p−1 2 ⌉.Proof.Let a graph G be eccentric colorable.Suppose on the contrary ω(G) > diam(G) then the vertices of maximum clique require at least diam(G) + 1 distinct colors, but eccentricity of every vertex of G is at most diam(G).Hence, G is not eccentric colorable, a contradiction.Hence, ω(G) ≤ diam(G).
Theorem 2.2.If a graph G is eccentric colorable then G contains no K m,n , m, n ≥ diam(G) as its subgraph.
Proof.Let a graph G be eccentric colorable.Suppose on the contrary G contains K m,n , where m, n ≥ diam(G) as subgraph then the vertices of K m,n require at least diam(G) + 1 colors, but e(v) ≤ diam(G) for all v ∈ V(G), a contradiction to the fact that G is eccentric colorable.Hence, G contains no K m,n , m, n ≥ diam(G) as its subgraph.
Proof.For χ e (G) = 2, G must contain at least two vertices.The only connected graph on two vertices is K 2 , which is not eccentric colorable and every connected graph with at least three vertices contains a path P 3 which requires at least three colors.Hence, there exists no connected graph G with χ e (G) = 2.
Remark 2. Any disconnected graph is eccentric colorable, since the eccentricity of every vertex in G is infinite.

Eccentric Coloring of Cycle With Chord
In this section we obtain results on eccentric colorability of a cycle with a chord between two vertices at distance two and eccentric colorability of a cycle with a chord between two vertices at distance three.
If p = 4(n + 1), n ≥ 1, color the vertices of C p with the sequence 1 2 1 3, . . .till all the vertices are colored.The resulting coloring is eccentric coloring with χ e (C p ) = 3, since for any connected graph χ e (G) ≥ 3.If p 4(n + 1), n ≥ 1, color the vertices of C p with the sequence 1 2 1 3, . . .till p − 1 vertex and color the vertex v p with color 4. The resulting coloring is eccentric coloring with χ e (C p ) = 4, Lemma 3.1.A cycle C p , p ≥ 9 with a chord between two vertices at distance two from each other is eccentric colorable.