The Sigma Chromatic Number of Corona of Cycles or Paths with - PowerPoint PPT Presentation

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References The Sigma Chromatic Number of Corona of Cycles or Paths with Complete Graphs Agnes D. Garciano 1 Reginaldo M. Marcelo 1 Maria Czarina T. Lagura 2 Nelson R. Tumala, Jr. 3 Ateneo de Manila University 1 University of Santo Tomas - Senior High School 2 Ozamis City National High School 3 May 21, 2018

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Overview Preliminaries 1 Basic Concepts Known Results Corona Graphs C m ⊙ K n and P m ⊙ K n 2 Main Results 3 Open Problems 4 References 5

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Basic Concepts Let G be a simple connected graph. Definition (Vertex Coloring) A vertex coloring of G is a mapping c : V ( G ) − → N .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Basic Concepts Let G be a simple connected graph. Definition (Vertex Coloring) A vertex coloring of G is a mapping c : V ( G ) − → N . Definition (Color Sum) ∀ v ∈ V ( G ) , � σ ( v ) = c ( u ) . u ∈ N G ( v )

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Basic Concepts Definition (Sigma Coloring) c is a sigma coloring if σ ( u ) � = σ ( v ) , ∀ uv ∈ E ( G ) .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Basic Concepts Definition (Sigma Coloring) c is a sigma coloring if σ ( u ) � = σ ( v ) , ∀ uv ∈ E ( G ) . Definition (Sigma Chromatic Number) σ ( G ) is the least number of colors required in a sigma coloring.

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Example G :

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Example A sigma coloring of G : ( a < b and b � = 2 a ) b b a a a b

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Example A sigma coloring of G : ( a < b and b � = 2 a ) σ = 2 a + 2 b σ = 2 a + b b b σ = a + b σ = 2 b a a a b σ = a + 3 b σ = 2 a + b

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Example A sigma coloring of G : ( a < b and b � = 2 a ) σ = 2 a + 2 b σ = 2 a + b b b σ = a + b σ = 2 b a a a b σ = a + 3 b σ = 2 a + b Thus, σ ( G ) = 2.

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Example For instance, if a = 1 and b = 3: σ = 8 σ = 5 3 3 σ = 4 σ = 6 1 1 1 3 σ = 10 σ = 5

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Observation 1 If u and v are adjacent vertices where N ( u ) − { v } = N ( v ) − { u } , then σ ( u ) � = σ ( v ) if and only if c ( u ) � = c ( v ) . u v

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Observation 1 If u and v are adjacent vertices where N ( u ) − { v } = N ( v ) − { u } , then σ ( u ) � = σ ( v ) if and only if c ( u ) � = c ( v ) . u v

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Observation 1 If u and v are adjacent vertices where N ( u ) − { v } = N ( v ) − { u } , then σ ( u ) � = σ ( v ) if and only if c ( u ) � = c ( v ) .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Theorem 2 (Chartrand, Okamoto, Zhang [2]) For any graph G , σ ( G ) ≤ χ ( G ) .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Theorem 2 (Chartrand, Okamoto, Zhang [2]) For any graph G , σ ( G ) ≤ χ ( G ) . Corollary 3 For the complete graph K n , σ ( K n ) = n .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Corollary 4 � 1 , if m = 1 or m = 3 , If m is any positive integer then σ ( P m ) = 2 , if m �∈ { 1 , 3 } .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Corollary 4 � 1 , if m = 1 or m = 3 , If m is any positive integer then σ ( P m ) = 2 , if m �∈ { 1 , 3 } . Corollary 5 � 2 , if m is even, For every integer m ≥ 3 , σ ( C m ) = 3 , if m is odd.

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Another way to represent color sums: (1 , 2) (2 , 1) σ = a + 2 b σ = 2 a + b b b (0 , 2) a σ = 2 b a b σ = 3 b σ = 2 a + b (0 , 3) (2 , 1)

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Lemma 6 Let { ( α 1 , β 1 ) , ( α 2 , β 2 ) , . . . , ( α r , β r ) } be a finite set of distinct or- dered pairs of nonnegative integers. Then there exist positive inte- gers a and b where a < b such that α i · a + β i · b � = α j · a + β j · b , for i � = j and 1 ≤ i , j ≤ r .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Known Results Lemma 7 Let { ( α 1 , β 1 , γ 1 ) , ( α 2 , β 2 , γ 2 ) , . . . , ( α r , β r , γ r ) } be a finite set of dis- tinct ordered triples of nonnegative integers. Then there exist posi- tive integers a , b and d where a < b < d such that α i · a + β i · b + γ i · d � = α j · a + β j · b + γ j · d , for i � = j and 1 ≤ i , j ≤ r .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Definition The corona of two graphs G and H , written as G ⊙ H , is the graph obtained by taking one copy of G and | V ( G ) | copies of H , where the i th vertex of G is adjacent to every vertex in the i th copy of H . G ⊙ H : H H H G H

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Definition Corona Graph C m ⊙ K n The corona graph C m ⊙ K n is the graph obtained by taking one copy of C m C m ⊙ K 4 : . . .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Definition Corona Graph C m ⊙ K n The corona graph C m ⊙ K n is the graph obtained by taking one copy of C m and m copies of K n C m ⊙ K 4 : . . . . . .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Definition Corona Graph C m ⊙ K n The corona graph C m ⊙ K n is the graph obtained by taking one copy of C m and m copies of K n where the i th vertex of C m is adjacent to every vertex in the i th copy of K n . C m ⊙ K 4 : . . . . . .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Corona Graph P m ⊙ K n P m ⊙ K 5 : . . . . . .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Preliminary Observation 9 Let G and H be disjoint graphs with sigma colorings c 1 and c 2 , respectively. Define c as the coloring of G ⊙ H given by � c 1 ( v ) , if v ∈ V ( G ) c ( v ) = c 2 ( v ) , if v is a vertex in any copy of H in G ⊙ H .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Preliminary Observation 9 Let G and H be disjoint graphs with sigma colorings c 1 and c 2 , respectively. Define c as the coloring of G ⊙ H given by � c 1 ( v ) , if v ∈ V ( G ) c ( v ) = c 2 ( v ) , if v is a vertex in any copy of H in G ⊙ H . If u and v are adjacent vertices that are both in G or both in H , then σ c ( u ) � = σ c ( v ) .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Main Result Theorem 10 Let m and n be positive integers with m ≥ 2 and n ≥ 2. Then, σ ( P m ⊙ K n ) = n .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Outline of the Proof: We want to show that σ ( P m ⊙ K n ) = n .

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Outline of the Proof: Let P m : c 1 : a sigma 2-coloring of P m , . . . a b a

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Outline of the Proof: Let K 5 : c 1 : a sigma 2-coloring of P m , c 2 : a sigma n -coloring of K n a e a b f a . . . with c 1 ( V ( P m )) ⊆ c 2 ( V ( K n )) . e f d f d d e b b Sigma coloring of K 5

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Outline of the Proof: Let P m ⊙ K 5 : c 1 : a sigma 2-coloring of P m , c 2 : a sigma n -coloring of K n a e a b f a . . . with c 1 ( V ( P m )) ⊆ c 2 ( V ( K n )) . e f d f d d e b b Let c be the coloring of P m ⊙ K n obtained from c 1 and c 2 . . . . a a b

Preliminaries Corona Graphs C m ⊙ K n and P m ⊙ K n Main Results Open Problems References Outline of the Proof: Note that P m ⊙ K 5 : σ ( P m ⊙ K n ) ≥ n a e a b f a . . . e f d f d d since the restriction of a sigma e b b coloring to the subgraph K n must also be a sigma coloring. . . . σ ( P m ⊙ K 5 ) ≥ 5 .