Adjacent Vertex Distinguishing Definitions Some Known Results - PowerPoint PPT Presentation

Adjacent Vertex Distinguishing Definitions Some Known Results Colorings of Graphs Main Results Outline of Proofs Home Page Title Page ◭◭ ◮◮ Wang Weifan ◭ ◮ Page. 1 Total 80 Department of Mathematics Back To Full Screen Zhejiang Normal University Close Jinhua 321004 Exit

� Definitions Definitions Some Known Results Main Results Outline of Proofs � Some Known Results Home Page Title Page ◭◭ ◮◮ � Our Main Results ◭ ◮ Page. 2 Total 80 Back To � Outline of Proofs Full Screen Close Exit

1 Definitions ¶ Let G = ( V, E ) be a simple graph. If G is a plane Definitions Some Known Results graph, let F denote the set of faces in G . Main Results Outline of Proofs ¶ Edge- k -coloring: Home Page Title Page A mapping f : E → { 1 , 2 , . . . , k } such that ◭◭ ◮◮ f ( e ) � = f ( e ′ ) for any adjacent edges e, e ′ ∈ E . ◭ ◮ Page. 3 Total 80 Back To ¶ Edge chromatic number: Full Screen χ ′ ( G ) = min { k | G is edge- k -colorable } . Close Exit

¶ Total- k -coloring: Definitions A mapping f : V ∪ E → { 1 , 2 , . . . , k } such that Some Known Results Main Results any two adjacent vertices, adjacent edges, and Outline of Proofs incident vertex and edge are assigned to different colors. Home Page Title Page ◭◭ ◮◮ ¶ Total chromatic number: ◭ ◮ Page. 4 Total 80 χ ′′ ( G ) = min { k | G is total- k -colorable } . Back To Full Screen Close Exit

¶ For an edge coloring f of G and for a vertex Definitions v ∈ V , we define: Some Known Results Main Results Outline of Proofs C f ( v ) = { f ( e ) | e is incident to v } . Home Page Title Page ¶ For a total coloring f of G and for a vertex ◭◭ ◮◮ v ∈ V , we define: ◭ ◮ Page. 5 Total 80 Back To C f [ v ] = { f ( e ) | e is incident to v } ∪ { f ( v ) } . Full Screen Close Exit

¶ Vertex-Distinguishing edge coloring (VD edge coloring) or Strong edge coloring: Definitions Some Known Results A proper edge coloring f such that Main Results Outline of Proofs C f ( u ) � = C f ( v ) Home Page for any two vertices u, v ∈ V . Title Page ◭◭ ◮◮ ◭ ◮ ¶ Vertex-Distinguishing edge chromatic number Page. 6 Total 80 (VD edge chromatic number): Back To Full Screen χ ′ s ( G ) = min { k | G is VD edge- k -colorable } . Close Exit

¶ Vertex-Distinguishing total coloring (VD total coloring) or Strong total coloring:: Definitions Some Known Results A proper total coloring f such that Main Results Outline of Proofs C f [ u ] � = C f [ v ] Home Page for any two vertices u, v ∈ V . Title Page ◭◭ ◮◮ ◭ ◮ ¶ Vertex-Distinguishing total chromatic number Page. 7 Total 80 (VD edge chromatic number): Back To Full Screen χ ′′ s ( G ) = min { k | G is VD total- k -colorable } . Close Exit

¶ Adjacent-Vertex-Distinguishing edge coloring (AVD edge coloring): Definitions Some Known Results A proper edge coloring f such that Main Results Outline of Proofs C f ( u ) � = C f ( v ) Home Page for any adjacent vertices u, v ∈ V . Title Page ◭◭ ◮◮ ◭ ◮ ¶ Adjacent-Vertex-Distinguishing edge chromatic Page. 8 Total 80 number (AVD edge chromatic number): Back To Full Screen χ ′ a ( G ) = min { k | G is AVD edge- k -colorable } . Close Exit

¶ Adjacent-Vertex-Distinguishing total coloring (AVD total coloring): Definitions Some Known Results Main Results A proper total coloring f such that Outline of Proofs C f [ u ] � = C f [ v ] Home Page for any adjacent vertices u, v ∈ V . Title Page ◭◭ ◮◮ ¶ Adjacent-Vertex-Distinguishing total chromatic ◭ ◮ Page. 9 Total 80 number (AVD edge total chromatic number): Back To χ ′′ a ( G ) = min { k | G is AVD total- k -colorable } . Full Screen Close Exit

Examples Definitions Some Known Results First Example: C 5 Main Results Outline of Proofs χ ′ ( C 5 ) = 3 , Home Page Title Page ◭◭ ◮◮ χ ′ a ( C 5 ) = 5 , ◭ ◮ Page. 10 Total 80 Back To χ ′′ a ( C 5 ) = χ ′′ ( C 5 ) = 4 . Full Screen Close Exit

1 Definitions Some Known Results 2 2 1 3 Main Results Outline of Proofs 4 4 Home Page 5 3 1 1 Title Page ◭◭ ◮◮ 2 3 4 4 ◭ ◮ χ = ' ( ) 5 χ = a C " Page. 11 Total 80 a C ( ) 4 5 5 Back To Full Screen Close Exit

Second Example: K 4 Definitions Some Known Results Main Results Outline of Proofs χ ′ ( K 4 ) = 3 , Home Page Title Page χ ′ a ( K 4 ) = 5 , ◭◭ ◮◮ ◭ ◮ Page. 12 Total 80 χ ′′ a ( K 4 ) = χ ′′ ( K 4 ) = 5 . Back To Full Screen Close Exit

1 Definitions Some Known Results Main Results Outline of Proofs 5 4 2 4 1 3 2 Home Page 3 1 1 5 Title Page ◭◭ ◮◮ 3 4 5 ◭ ◮ 2 χ = Page. 13 Total 80 χ = " ' ( ) 5 a K a K ( ) 5 4 4 Back To Full Screen Close Exit

2 Some Known Results Definitions Some Known Results Main Results Outline of Proofs ∆ : the maximum degree of a graph G Home Page δ : the minimum degree of a graph G Title Page d -Vertex: a vertex of degree d ◭◭ ◮◮ ◭ ◮ n : the number of vertices of a graph G Page. 14 Total 80 Back To n d : the number of d -vertices in G Full Screen Close Exit

(2.1) Strong Edge Coloring (VD Edge Coloring) The concept of strong edge coloring was intro- Definitions Some Known Results duced independently by Aigner, Triesch, and Tuza, Main Results Outline of Proofs by Horn ´ a k and Sot ´ a k, and by Burris and Schelp. Home Page Title Page A graph G has a strong edge coloring if and only if G contains no isolated edges, and G has at most one isolated vertex. In this part, we assume that ◭◭ ◮◮ G has no isolated edges and has at most one isolated vertex. ◭ ◮ Page. 15 Total 80 Combinatorial degree µ ( G ) : Back To � k Full Screen � µ ( G ) = max δ ≤ d ≤ ∆ min { k | ≥ n d } . d Close Exit

Definitions Conjecture 2.1 (Burris and Schelp, 1997) Some Known Results Main Results Outline of Proofs For a graph G , µ ( G ) ≤ χ ′ s ( G ) ≤ µ ( G ) + 1 . Home Page Conjecture 2.2 (Burris and Schelp, 1997) Title Page ◭◭ ◮◮ For a graph G , χ ′ s ( G ) ≤ n + 1 . ◭ ◮ Page. 16 Total 80 [A.C.Burris, R.H.Schelp, J.Graph Theory, 26(1997) 73-82.] Back To Full Screen Close Exit

Theorem 2.1.1 For n ≥ 3 , χ ′ s ( K n ) = n if n is odd, Definitions Some Known Results and χ ′ s ( K n ) = n + 1 if n is even. Main Results Outline of Proofs Theorem 2.1.2 Let n be a cycle of length n ≥ 3 Home Page and let µ = µ ( C n ) . Then χ ′ s ( C n ) = µ + 1 if µ is Title Page � µ � µ � � − 2 ≤ n ≤ − 1 or µ is even and odd and ◭◭ ◮◮ 2 2 n > ( µ 2 − 2 µ ) / 2 , and χ ′ ◭ ◮ s ( C n ) = µ otherwise. Page. 17 Total 80 Back To [A.C.Burris and R.H.Schelp, J.Graph Theory, 26(1997) 73-82.] Full Screen Close Exit

Theorem 2.1.3 For a graph G , m 1 ≤ χ ′ s ( G ) ≤ (∆+ 1) ⌊ 2 m 2 + 5 ⌋ , where Definitions Some Known Results 1 Main Results k + k − 1 m 1 = 1 ≤ k ≤ ∆ { ( k ! n k ) max 2 } , Outline of Proofs 1 Home Page k m 2 = 1 ≤ k ≤ ∆ n max k . Title Page ◭◭ ◮◮ Corollary 2.1.4 If G is an r -regular graph of order ◭ ◮ 1 Page. 18 Total 80 n , then χ ′ r + 5 ⌋ . s ( G ) ≤ ( r + 1) ⌊ 2 n Back To Full Screen [A.C.Burris,R.H.Schelp, J.Graph Theory, 26(1997) 73-82.] Close Exit

Definitions Some Known Results Let c be the smallest number in the interval (4 , 6 . 35) such that Main Results Outline of Proofs 1 1 6 c 2 + c (49 c 2 − 208) 2 < max { n 1 + 1 , ⌈ cn 2 ⌉ , 21 } . 2 c 2 − 16 Home Page Theorem 2.1.5 For a tree T � = K 2 , we have Title Page 1 χ ′ 2 s ( T ) ≤ max { n 1 + 1 , ⌈ cn 2 ⌉ , 21 } . ◭◭ ◮◮ ◭ ◮ [A.C.Burris, R.H.Schelp, J.Graph Theory, 26(1997) 73-82.] Page. 19 Total 80 Back To Full Screen Close Exit

Theorem 2.1.6 Let G be a vertex-disjoint union of cycles, and let k be the least number such that n ≤ � k . Then χ ′ � s ( G ) = k or k + 1 . 2 Definitions Some Known Results Theorem 2.1.7 Let G be a vertex-disjoint union of Main Results Outline of Proofs paths with each path of length at least two. Let k be � k � the least number such that n 1 ≤ k and n 2 ≤ . 2 Home Page Then χ ′ s ( G ) = k or k + 1 . Title Page ◭◭ ◮◮ Theorem 2.1.8 Let G be a strong edge colorable ◭ ◮ graph with ∆ = 2 . Let k be the least number such Page. 20 Total 80 � k . Then k ≤ χ ′ � that n 1 ≤ k and n 2 ≤ s ( G ) ≤ Back To 2 Full Screen k + 5 . Close Exit [P.N.Balister, B.Bollb ´ a s, R.H.Schelp, Discrete Math., 252(2002) 17-29.]

Theorem 2.1.9 If L m is an m -sided prism, then χ ′ s ( L m ) ≤ µ ( L m ) + 1 . Definitions Some Known Results Main Results Outline of Proofs Let G be a graph and r ≥ 1 be an integer. Let rG denote the graph obtained from G by replacing Home Page each edge of G with r multi-edges. Title Page ◭◭ ◮◮ Theorem 2.1.10 Let r ≥ 1 be an integer. Then ◭ ◮ χ ′ s ( rK 4 ) ≤ µ ( rK 4 ) + 1 . Page. 21 Total 80 Back To Full Screen [K.Taczuk, M.Wo ´ z niak, Opuscula Math., 24/2(2004) 223-229.] Close Exit