Prime Coloring of Crossing Number Zero Graphs

Received: February 11, 2019 Revised: April 08, 2019 Accepted: May 07, 2019 Published online: September 11, 2019 In this paper, prime coloring and its chromatic number of some crossing number zero graphs are depicted and its results are vali-dated with few theorems. Prime Coloring is defined as G be a loop less and Without multiple edges with n distinct Vertices on Color class C={c1,c2,c3,.....cn} a bijection ψ :V {c1,c2,c3,.....cn} if for each edge e = cicj, i≠j , gcd{ ψ (ci), ψ (cj)}=1, ψ (ci) and ψ (cj) receive distinct Colors. The Chromatic number of Prime coloring is minimum cardinality taken by all the Prime colors. It is denoted by η( ). G


Introduction
We use the terminology as Bondy.J.A. and Murty U.S.R (1976). Let G= (V, E) be simple graph with crossing number zero i.e (Cr(G)=0) and here order of V be| V |= n. In this paper, we discuss Prime graphs and Coloring. The Prime labeling emanate with Entringer et al. 1980. Be the graph G = (V, E) with vertex set V is said to have a Prime labeling if there exists a bijection f: V(G) →Z + -{0}, Z + is the set of positive integers such that for each edge xy E G ∈ ( ) ,gcd {f(x),f(y)}=1. It begin to be an interesting problem to investigate the Prime coloring as possible in a graph with n vertices. Moreover, the general graph G is proper colorable but not prime graph forever. The Coloring of a graph is explained as pair wise adjacent vertices receive conflict colors.

Prime Coloring of Crossing Number Zero Graphs
In this entire paper, we consider ψ is the bijective function and ′ ψ is the Chromatic number of G as well as η is also chromatic number of prime graph.
Theorem 2.1 Every path P n graph is Prime if it is Prime colorable graph.

Proof:
Let the path P n with n distinct vertices, the bijection ψ :V(P n )→{ c 1 ,c 2 ,c 3 ,…..c n } graph if for each edge e = c i c j , gcd{ ψ (c i ), ψ (c j )}=1, i≠j, ψ (c i ) =c i ψ (c j )=c j , receive distinct colors. Then the path P n Graph is Prime colorable graph.
Corollary 2.1.1: If path P n graph is prime then η( ) . P n = 2 Example 2.1: P n is Prime colorable graph Theorem 2.2: Every Cycle C n , n≥4 if it is Prime colorable graph.
Proof: Assume that Cycle C n on n≥4 vertices with the bijection ψ :V(C n ) → {c 1 ,c 2 ,c 3 ,…..c n } graph if for each edge e = c i c j , gcd{ψ (c i ),ψ (c j )}=1,i≠j , ψ (c i ) =c i ψ (c j )=c j receive distinct colors on cycle C n on n≥4 .  Theorem 2.3 If a Complete graph K n , n≤3 is Prime graph, then K n is Prime colorable graph with n≤3.

Proof
Proof by induction n=1 obviously holds. n=2 then the graph is a path by the theorem (2.1) holds. n=3 then the graph is triangle every gcd {ψ (c i ), ψ (c j )} = 1, i≠j then the graph is prime and also receive distinct colors. n=4 then the vertices assigns distinct colors but gcd {ψ (c i ), ψ (c j )} ≠ 1. But e = c i c j, then the graph is not a prime. Therefore n>3, K n is not a prime colorable graph. Corollary 2.3.1 Every Complete graph K n is Proper Colorable but not Prime colorable. Consider K n is complete graph pair wise adjacent vertices are connected by the definition of graph coloring it has to assign distinct colors, but the labeled graph K n with {1,2,….n} ,gcd (1,n) not equal to 1 for n ≥3, K n is not Prime graph. Thus K n is not a Prime colorable

Example 2.3
The complete graph K 3 is Prime Cloring graph.
Complete graph K 4

Proof
Let V={c 1 ,c 2 ,c 3 …….c n+1 } vertices of Star graph K 1,n . The graph labeled as well as colored, by the bijection ψ :V(K 1,n ) → { c 1 ,c 2 ,c 3 ,…..c n+1 } graph if for each edge e= c i c j ,gcd{ ψ (c i ), ψ (c j )}=1, i≠j ψ (c i ) and ψ (c j ) receive distinct colors .Therefore the coloring pairs of are the distinct. The coloring pairs are at most 2 and also every gcd (i,j)=1 is a prime graph. Thus, ψ (c i )=c i ψ (c j )=c j , for 1≤i,j≤n+1 then every i≠j is Prime colorable. Therefore minimum coloring needed for the Star graph is η( ) . Theorem 2.5 If Wheel graph W n , n≥4 is Prime graph, then η( ) . W n = 4 when n is even.

Proof
Let V= {c 1 ,c 2 ,c 3 …….c k+1 } vertices of wheel graph W n . The graph labeled as well as colored. By the bijection ψ :V(W n ) → { c 1 ,c 2 ,c 3 ,…..c k+1 }graph if for each edge e = c i c j , gcd{ ψ (c i ),(c j )}=1, i≠j ,ψ (c i ) and ψ (c j ) receive distinct colors. Therefore the coloring pairs of are the distinct. Therefore the coloring pairs are at most four and also gcd(i,j)=1 is a prime graph. Thus,ψ (c i )=c i ψ (c j )=c j for 1≤i,j≤k+1 Then every i≠j is prime colorable. Therefore minimum coloring needed for the Wheel graph is η( ) . , K n 1 2 =

Main Results
Lemma 3.1: G is Prime graph ⇔ G is Prime colorable graph.

Proof
Let us assume G is prime label with the bijection ψ :V(G) → { c 1 ,c 2 ,c 3 ,…..c n }graph if for each edge w = c i c j , gcd {ψ (c i ), ψ (c j )}=1 , i≠j ,ψ (c i ) & ψ (c j ) receives distinct colors. Now coloring the vertices is proper coloring. Then the resulting graph with different colors. Conversely G is Prime coloring by the definition of graph coloring holds it is a prime graph.
Corollary 3.1.2 Every spanning sub graph of prime graph is prime colorable graph. Seoud and yousse et.al (1999) proved every spanning sub graph is prime graph. Now coloring proper by the graph become an prime colorable gcd{ ψ (c i ), ψ (c j )}=1, i≠j, ψ (c i ) and ψ (c j ) receive dis tinct colors. We observe that every spanning sub graph is also prime colorable.
Corollary 3.1.3: Every tree is prime colorable graph. Every tree is prime is proved by Entringer n≤50 vertices . The tree have been properly colored as the color class C={ c 1 ,c 2 ,c 3 ,…..c n }. Tree is a spanning sub graph of the general graph G by the above corollary (3.1.1) holds, there for every tree is prime colorable. i. P n is Path. ii. P n is Prime graph. iii. P n is Prime colorable graph.

Proof
(i) ⇒ (ii) let the path P n with n distinct vertices then the chromatic number of the graph ′ = ψ ( ) P n 2 we have to prove P n is a prime graph with gcd{ ψ (c i ), ψ (c j )}=1, i≠j ,then P n is a prime graph. (ii) ⇒ (iii) By the theorem (2.1) refers to every prime graph P n which assigned the distinct colors and by the definition of a prime color then η( ) P n = 2 .
(iii) ⇒ (i) from the proof given prime colorable graph ′ = = ψ η ( ) ( ) P P n n 2 Then n distinct vertices are connected as a path.

Lemma 3.3
n is even n is odd , then the following are equivalent i. C n is Cycle ii. C n is Prime graph.
iii. C n is Prime colorable graph.

Proof
Case (i) n is odd: (i) ⇒ (ii) Consider C n be the cycle with distinct n vertices. If Cycle C n , n>3 is an odd cycle with 2n-1 vertices (Samir K.Vaidya, Udayan M. Prajapati) (2012) et. al., by the Fusion, switching, duplication of the vertices than the graph is the prime graph.
(ii) ⇒ (iii) Consider C n is prime graph when n is odd, be the fusion, switching, duplication is allowed then the graph is prime by the definition of prime coloring C n is prime colorable graph. = ≠ -,i j vertices are distinct and even vertices of C n is cycle. Lemma 3.4 If G is the prime graph and ∆ is the maximum degree of the graph then η( ) G ≤ + ∆ 1

Conclusion
This article discussed & absorbed the upshot of graph coloring technique which associated with the prime labeled graph. We elucidated the coloring techniques of some crossing number zero graphs and also evinced the some interesting results obtained by Prime Coloring graphs.