Relation between Center Coloring and the other Colorings

In this paper, center coloring and center coloring number are defined, some bounds are established for the center coloring number of a graph in terms of other graphical coloring parameters, and a polynomial time algorithm is proposed in order to calculate the center coloring of a graph.


Introduction
Over the 150 years, various works have been done on the coloring of graphs such as vertex coloring, edge coloring and etc.A coloring of a graph G is an assignment of colors to the vertices of G, one color to each vertex, so that adjacent vertices are assigned different colors.A coloring in which k colors are used is a k-coloring.The minimum integer k for which a graph G is k-colorable is called the chromatic number of G and is denoted by ( ) G  (Chartrand et. al., 2009).
An assignment of colors to the edges of a nonempty graph G so that adjacent edges are colored differently is an edge coloring of G.The graph G is k-edge colorable if there exists an edge coloring of G for some   k.The minimum integer k for which a graph G is k-edge colorable is its edge chromatic number and is denoted by 1 ( ) G  (Chartrand et. al., 2009).
A total coloring of a graph G is an assignment of colors to the elements (vertices and edges) of G so that adjacent elements and incident elements of G are colored differently.A k total coloring is a total coloring that uses k colors.The minimum k for which a graph G admits a k-total coloring is called the total coloring number of G and is denoted by 2 ( ) G  (Chartrand et. al., 2009).
A harmonious coloring of a simple graph G is proper vertex coloring such that each pair of colors appears together on at most one edge.Formally, a harmonious coloring is a function from a color set to the set (G) of vertices of G such that for any edge e of G with end points , say ( ) ≠ ( ), and for any pair of distinct edges , ′ with end points , and , ′ respectively, then { ( ), ( )} ≠ { ( ), ( )}.The harmonious chromatic number (G) is the least number of colors in such a coloring (Chartrand & Lesniak, 2005).
For a nontrivial connected graph G, let : ( ) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same.For a vertex of G, the neighborhood color set ( ) is the set of colors of the neighbors of .The coloring is called a set coloring if ( ) ≠ ( ) for every pair , of adjacent vertices of G.The minimum number of colors required of such a coloring is called the set chromatic number ( ) of G (Chartrand et. al., 2009).
Let G be a simple, connected graph with n vertices and m edges.We define a k-coloring of a graph as a mapping f from the vertices of G onto the set {1, 2, ..., k}.Let be an edge between vertices u and v.If u and v are assigned colors f (u) and f (v) respectively, then the color of is defined by f ( ) = {f (u), f (v)}.A line-distinguishing coloring of G is a k-coloring of G such that no two edges have the same color.In other words, if and are any two edges in G, then f ( ) f ( ).Note that it is not required that each allowable pair of colors appears exactly once.The line-distinguishing chromatic number ( ) G  is defined as the smallest k such that G has a linedistinguishing k-coloring.Note that two adjacent vertices may have the same color (Immelman, 2007).
Let G be a nontrivial connected graph on which an edge-coloring c: E (G)→ {1,2, • • • ,n}, n ∈ ℕ, is defined, where adjacent edges may be colored the same.A path is rainbow if no two edges of it are colored the same.An edge-colored graph G is rainbow connected if every two distinct vertices are connected by a rainbow path.An edge-coloring under which G is rainbow connected is called a rainbow coloring.Clearly, if a graph is rainbow connected, it must be connected.Conversely, every connected graph has a trivial edge-coloring that makes it rainbow connected by coloring edges with distinct colors.Thus, we define the rainbow connection number of a connected graph G, denoted by rc (G), as the smallest number of colors that are needed in order to make G rainbow connected.A rainbow coloring using rc (G) colors is called a minimum rainbow coloring (Li & Sun, 2012).
A vertex-colored graph G is rainbow vertex-connected if its every two distinct vertices are connected by a path whose internal vertices have distinct colors.A vertex-coloring under which G is rainbow vertex-connected is called a rainbow vertex-coloring.The rainbow vertex-connection number of a connected graph G, denoted by rvc (G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected (Li & Sun, 2012).A center coloring of a graph is an assignment of colors to the vertices of G, one color to each vertex so that different distance vertices from the center are assigned different colors.Two adjacent vertices can receive the same color.The number of colors required of such a coloring is called center coloring number Cc (G) of G (Yorgancıoğlu et. al., 2015).This coloring can be applied to hierarchy problems to find the number of structures, people, criteria and comparisons, etc. Moreover it can be applied to earthquake motion problems to find the number of settlements that are affected by an earthquake.
The distance d (u, v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G (Buckley & Harary, 1990).
The eccentricity e (v) of a vertex v in a connected graph G is the distance from v to a vertex farthest from v in G (Buckley & Harary, 1990).
The radius rad (G) of a connected graph G is defined as the minimum eccentricity among the vertices of G and the diameter diam (G) is a maximum eccentricity among the vertices of G (Buckley & Harary, 1990).And v is a central vertex if e (v) = rad (v) and the center C (G) is the set of all central vertices (Buckley & Harary, 1990).Some graphs G have the property that each vertex of G is a central vertex.A graph is selfcentered if every vertex is in the center (Buckley & Harary, 1990).

Theorem 1.2. Vizing Theorem
If G is a nonempty graph, then, Theorem 1.3.
( )= n for any graph G of a diameter at most 2 (Miller & Pritikin, 1991).

Some Bounds For Center Coloring
In this section, we give some bounds for center coloring number for graphs.
Theorem 2.1.If G is a connected graph with n vertices that is not a tree, then .
Proof: If the graph is self-centered graph, its center coloring number is 1.To prove the right side of the inequality, the distance from the center vertex of the graph to the furthest vertices is . So one more color is added for the center coloring to get 1 2 If G is a connected graph that is not a tree and not a self-centered graph, then, for n Proof: The vertex degree of graph is at most is n-1.From theorem 2.1, , is obtained.To prove the right side of the inequality, the distance between the center and other vertices can be at least 1, so the center coloring number is at least 2.
Theorem 2.3.If G and G is a connected graph of order n, then, Proof: First we verify the upper bound for ( G ) + ( G ) and ( G ). ( G ).
( ) (Yorgancıoğlu et. al., 2015) but for the connected graph G , ( ) Since the arithmetic mean of two positive numbers is always at least as large as their geometric mean, we have To verify the lower bound for (i) and (ii) G and G graphs can not be self-centered graphs so center coloring number of G is at least 2.
So 2+2  ( G )+ ( G ) and 2.2  ( G ) ( G ). C T a  and let there be given set a-coloring of G using the colors in ℕ .Since there are ( 2 1 a  ) non-empty subsets of ℕ and total coloring number is at most ( 2 1 a  ). So, To verify the lower bound;

Relations Between Center Coloring and the other colorings
In this section, we compare the center coloring number with some other coloring numbers.
Theorem 3.1.If is a connected graph, then .
Proof: Since (theorem 2.2) and (Immelman, 2007) are given, the inequality follows.Then the inequality is clear.
Proof: In the definition of set chromatic number, the neighborhood color sets of each adjacent vertex must be given differently.But in center coloring adjacent vertices may have same color set and adjacent vertices may have the same color.So it is clear from the definition that center coloring number is smaller than the set chromatic number.■ Theorem 3.3.If G is a connected graph that is not a tree, then ( ) Chartrand et al., 2009) and in theorem 3.2 ( ) ( ) Also in M. Kubale (Kubale, 2004) " For any graph G with diameter at most 2, ( ) h G n


 " is given and for any connected graph with diameter at most 2, the center coloring number of these graphs is at most 3.So it is obvious that center coloring number is smaller than the harmonious chromatic number for any connected graph with diameter at most 2. Yorgancıoğlu et al., 2015)] is given and also ( ) ■ Theorem 3.6.If G is a connected graph, that is not a star or a complete graph, then for n>4 ( ) Theorem 1.1.Brooks' Theorem Let G be a connected simple graph whose maximum vertex degree is d .If G is neither a cycle graph with an odd number of vertices, nor a complete graph, then ( diameter spanning tree is "a" and total coloring number is "b".Proof: To verify the upper bound; let ( ) c MDST is Floyd-Warshall which returns the distances matrix from adjacency matrix of graph G. Function Floyd-Warshall (G); Begin if i=j then wij ← 0 ; if vi disjoint vj then wij ←  ; for k =1 to n do for i=1 to n do for j=1 to n do { for j=1 to i do } if wij > wik + wkj then wij ← wik + wkj ;