Fractional Colorings and Zykov Products of graphs Who? Nichole - PowerPoint PPT Presentation

Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011

Graphs A graph , G , consists of a vertex set, V ( G ), and an edge set , E ( G ). V ( G ) is any finite set E ( G ) is a set of unordered pairs of vertices

Graphs A graph , G , consists of a vertex set, V ( G ), and an edge set , E ( G ). V ( G ) is any finite set E ( G ) is a set of unordered pairs of vertices Example Figure: Peterson graph

Subgraphs A subgraph H of a graph G is a graph such that V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ).

Subgraphs A subgraph H of a graph G is a graph such that V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). Example Figure: Subgraph of the Peterson graph

Subgraphs An induced subgraph , H , of G is a subgraph with property that any two vertices are adjacent in H if and only if they are adjacent in G .

Subgraphs An induced subgraph , H , of G is a subgraph with property that any two vertices are adjacent in H if and only if they are adjacent in G . Example Figure: Induced Subgraph of Peterson graph

Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges.

Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges. Example Figure: Independent set

Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges. Example Figure: Independent set The set of all independent sets of a graph G is denoted I ( G ).

Weighting I ( S ) A weighting of I ( G ) is a function w : I ( G ) → R ≥ 0 .

Weighting I ( S ) A weighting of I ( G ) is a function w : I ( G ) → R ≥ 0 . Example S w ( S ) { a } 1 / 3 { b } 1 / 3 a { c } 1 / 3 { d } 1 / 3 { e } 1 / 3 e b { a,c } 1 / 3 { a,d } 1 / 3 { b,d } 1 / 3 { b,e } 1 / 3 c d { e,c } 1 / 3 Figure: C 5 and a corresponding weighting

Fractional k -coloring A fractional k -coloring of a graph, G , is a weighting of I ( G ) such that � S ∈ I ( G ) w ( S ) = k ; and

Fractional k -coloring A fractional k -coloring of a graph, G , is a weighting of I ( G ) such that � S ∈ I ( G ) w ( S ) = k ; and For every v ∈ V ( G ), � w ( S ) = w [ v ] ≥ 1 S ∈ I ( G ) v ∈ S

Fractional k -coloring Example S w ( S ) { a } 1 / 3 a { b } 1 / 3 { c } 1 / 3 { d } 1 / 3 e { e } 1 / 3 b { a,c } 1 / 3 { a,d } 1 / 3 { b,d } 1 / 3 { b,e } 1 / 3 { e,c } 1 / 3 c d Figure: A fractional coloring of C 5 with weight 10 / 3

Fractional k -coloring Example S w ( S ) { a } 1 / 3 a { b } 1 / 3 { c } 1 / 3 { d } 1 / 3 e { e } 1 / 3 b { a,c } 1 / 3 { a,d } 1 / 3 { b,d } 1 / 3 { b,e } 1 / 3 { e,c } 1 / 3 c d Figure: A fractional coloring of C 5 with weight 10 / 3 � S ∈ I ( G ) w ( S ) = 10 / 3

Fractional k -coloring Example S w ( S ) { a } 1 / 3 a { b } 1 / 3 { c } 1 / 3 { d } 1 / 3 e { e } 1 / 3 b { a,c } 1 / 3 { a,d } 1 / 3 { b,d } 1 / 3 { b,e } 1 / 3 { e,c } 1 / 3 c d Figure: A fractional coloring of C 5 with weight 10 / 3 � S ∈ I ( G ) w ( S ) = 10 / 3 w [ v ] = 1 for every v ∈ V ( G )

Fractional Chromatic Number The fractional chromatic number , χ f ( G ), is the minimum possible weight of a fractional coloring.

Fractional Chromatic Number The fractional chromatic number , χ f ( G ), is the minimum possible weight of a fractional coloring. Example w ( S ) S { a } 0 a { b } 0 { c } 0 { d } 0 e { e } 0 b { a,c } 1 / 2 { a,d } 1 / 2 { b,d } 1 / 2 { b,e } 1 / 2 { e,c } 1 / 2 c d Figure: A weighting C 5

Fractional Chromatic Number Example S w ( S ) { a } 0 a { b } 0 { c } 0 { d } 0 e { e } 0 b { a,c } 1 / 2 { a,d } 1 / 2 { b,d } 1 / 2 { b,e } 1 / 2 { e,c } 1 / 2 c d Figure: A fractional 5 / 2-coloring of C 5

Fractional Chromatic Number Example S w ( S ) { a } 0 a { b } 0 { c } 0 { d } 0 e { e } 0 b { a,c } 1 / 2 { a,d } 1 / 2 { b,d } 1 / 2 { b,e } 1 / 2 { e,c } 1 / 2 c d Figure: A fractional 5 / 2-coloring of C 5 � S ∈ I ( G ) w ( S ) = 5 / 2

Fractional Chromatic Number Example S w ( S ) { a } 0 a { b } 0 { c } 0 { d } 0 e { e } 0 b { a,c } 1 / 2 { a,d } 1 / 2 { b,d } 1 / 2 { b,e } 1 / 2 { e,c } 1 / 2 c d Figure: A fractional 5 / 2-coloring of C 5 � S ∈ I ( G ) w ( S ) = 5 / 2 w [ v ] = 1 for every v ∈ V ( G )

Fractional Chromatic Number How do we know what the minimum is?

Fractional Chromatic Number How do we know what the minimum is? Linear Programming

Fractional Chromatic Number How do we know what the minimum is? Linear Programming Formulas

Zykov Product of Graphs

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows.

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows. Take the disjoint union of G i

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows. Take the disjoint union of G i Example Figure: Drawings of P 2 and P 3

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows. Take the disjoint union of G i For each ( x 1 , . . . , x n ) ∈ V ( G 1 ) × V ( G 2 ) × . . . × V ( G n ) add a new vertex adjacent to the vertices { x 1 , . . . , x n }

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows. Take the disjoint union of G i For each ( x 1 , . . . , x n ) ∈ V ( G 1 ) × V ( G 2 ) × . . . × V ( G n ) add a new vertex adjacent to the vertices { x 1 , . . . , x n } Example Figure: Constructing Z ( P 2 , P 3 )

Zykov Product of Graphs The Zykov product Z ( G 1 , G 2 , . . . , G n ) of simple graphs G 1 , G 2 , . . . , G n is formed as follows. Take the disjoint union of G i For each ( x 1 , . . . , x n ) ∈ V ( G 1 ) × V ( G 2 ) × . . . × V ( G n ) add a new vertex adjacent to the vertices { x 1 , . . . , x n } Example Figure: Z ( P 2 , P 3 )

Zykov Graphs

Zykov Graphs The Zykov graphs , Z n , are formed as follows: Set Z 1 as a single vertex

Zykov Graphs The Zykov graphs , Z n , are formed as follows: Set Z 1 as a single vertex Define Z n := Z ( Z 1 , ..., Z n − 1 ) for all n ≥ 2 Figure: Drawing of Z 1

Zykov Graphs The Zykov graphs , Z n , are formed as follows: Set Z 1 as a single vertex Define Z n := Z ( Z 1 , ..., Z n − 1 ) for all n ≥ 2 Figure: Drawings of Z 1 and Z 2

Zykov Graphs The Zykov graphs , Z n , are formed as follows: Set Z 1 as a single vertex Define Z n := Z ( Z 1 , ..., Z n − 1 ) for all n ≥ 2 Figure: Drawings of Z 1 , Z 2 , and Z 3

Zykov Graphs The Zykov graphs , Z n , are formed as follows: Set Z 1 as a single vertex Define Z n := Z ( Z 1 , ..., Z n − 1 ) for all n ≥ 2 Figure: Drawing of Z 4

Jacobs’ Conjecture

Jacobs’ Conjecture Corollary For n ≥ 1 , 1 χ f ( Z n +1 ) = χ f ( Z n ) + χ f ( Z n )

Jacobs’ Conjecture Corollary For n ≥ 1 , 1 χ f ( Z n +1 ) = χ f ( Z n ) + χ f ( Z n ) Example χ f ( Z 1 ) = 1

Jacobs’ Conjecture Corollary For n ≥ 1 , 1 χ f ( Z n +1 ) = χ f ( Z n ) + χ f ( Z n ) Example χ f ( Z 1 ) = 1 χ f ( Z 2 ) = 1 + 1 1 = 2

Jacobs’ Conjecture Corollary For n ≥ 1 , 1 χ f ( Z n +1 ) = χ f ( Z n ) + χ f ( Z n ) Example χ f ( Z 1 ) = 1 χ f ( Z 2 ) = 1 + 1 1 = 2 χ f ( Z 3 ) = 2 + 1 2 = 5 2

Verifying χ f ( C 5 )

Verifying χ f ( C 5 ) Notice that → Figure: Z 3 and C 5

Verifying χ f ( C 5 ) Notice that → Figure: Z 3 and C 5 So, χ f ( Z 3 ) = χ f ( C 5 ) = 5 / 2

The Main Result: Theorem 1

The Main Result: Theorem 1 Theorem For n ≥ 2 , let G 1 , . . . , G n be graphs. Set G := Z ( G 1 , . . . , G n ) and χ i = χ f ( G i ) . Suppose also that the graphs G i are numbered such that χ i ≤ χ i +1 . Then n n � �� � 1 − 1 � � χ f ( G ) = max χ n , 2 + χ k i =2 k = i

The Main Result: Theorem 1 Theorem For n ≥ 2 , let G 1 , . . . , G n be graphs. Set G := Z ( G 1 , . . . , G n ) and χ i = χ f ( G i ) . Suppose also that the graphs G i are numbered such that χ i ≤ χ i +1 . Then n n � �� � 1 − 1 � � χ f ( G ) = max χ n , 2 + χ k i =2 k = i Example � � �� 1 − 1 χ f ( Z ( P 2 , P 3 )) = max 2 , 2 + 2 = max(2 , 5 2) = 5 2 .

Lower Bound: χ f ( G ) ≥ max ( χ n , f ( n ))

Lower Bound: χ f ( G ) ≥ max ( χ n , f ( n )) Lemma The fractional chromatic number of a subgraph, H, is at most equal to the fractional chromatic number of a graph, G.

Lower Bound: χ f ( G ) ≥ max ( χ n , f ( n )) Lemma The fractional chromatic number of a subgraph, H, is at most equal to the fractional chromatic number of a graph, G. Conclusion χ f ( G ) ≥ χ n

Lower Bound: χ f ( G ) ≥ max ( χ n , f ( n )) Lemma Let G be a graph and w a weighting of X ⊆ I ( G ) . Then, for every induced subgraph H of G, there exists x ∈ V ( H ) such that 1 � w [ x ] ≤ w ( S ) . χ f ( H ) S ∈ X

Lower Bound: χ f ( G ) ≥ max ( χ n , f ( n )) Start with w , a χ f -coloring of G and x 1 ∈ V ( G 1 ).