Rainbow Edge-coloring and Rainbow Domination Douglas B. West - PowerPoint PPT Presentation

Rainbow Edge-coloring and Rainbow Domination Douglas B. West Department of Mathematics University of Illinois at Urbana-Champaign west@math.uiuc.edu slides available on DBW preprint page Joint work with Timothy D. LeSaulnier

The Problem edge-coloring: cover E ( G ) with matchings — χ ′ ( G ) domination: cover V ( G ) with disjoint stars — γ ( G )

The Problem edge-coloring: cover E ( G ) with matchings — χ ′ ( G ) domination: cover V ( G ) with disjoint stars — γ ( G ) Def. rainbow subgraph: in an edge-colored graph, a subgraph whose edges have distinct colors

The Problem edge-coloring: cover E ( G ) with matchings — χ ′ ( G ) domination: cover V ( G ) with disjoint stars — γ ( G ) Def. rainbow subgraph: in an edge-colored graph, a subgraph whose edges have distinct colors Def. rainbow edge-coloring: use rainbow matchings χ ′ ( G ) = min { k : G has a rainbow k -edge-coloring} ˆ

The Problem edge-coloring: cover E ( G ) with matchings — χ ′ ( G ) domination: cover V ( G ) with disjoint stars — γ ( G ) Def. rainbow subgraph: in an edge-colored graph, a subgraph whose edges have distinct colors Def. rainbow edge-coloring: use rainbow matchings χ ′ ( G ) = min { k : G has a rainbow k -edge-coloring} ˆ Def. rainbow domination: use disjoint rainbow stars γ ( G ) = min { k : V ( G ) covered by k disjoint rainb. stars} ˆ

The Problem edge-coloring: cover E ( G ) with matchings — χ ′ ( G ) domination: cover V ( G ) with disjoint stars — γ ( G ) Def. rainbow subgraph: in an edge-colored graph, a subgraph whose edges have distinct colors Def. rainbow edge-coloring: use rainbow matchings χ ′ ( G ) = min { k : G has a rainbow k -edge-coloring} ˆ Def. rainbow domination: use disjoint rainbow stars γ ( G ) = min { k : V ( G ) covered by k disjoint rainb. stars} ˆ χ ′ ( G ) = χ ′ ( G ) . If the edge-coloring is rainbow, then ˆ If the edge-coloring is proper, then ˆ γ ( G ) = γ ( G ) .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column).

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings.

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ • ˆ α ′ ( K 4 ) = 1 when properly colored. Assume ˆ δ ( G ) ≥ 4 .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ • ˆ α ′ ( K 4 ) = 1 when properly colored. Assume ˆ δ ( G ) ≥ 4 . � 1 5 α ′ ( G ) ≥ 2 ˆ � Conj. (Wang–Li [2008]) ˆ δ ( G ) . They did 12 .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ • ˆ α ′ ( K 4 ) = 1 when properly colored. Assume ˆ δ ( G ) ≥ 4 . � 1 5 α ′ ( G ) ≥ 2 ˆ � Conj. (Wang–Li [2008]) ˆ δ ( G ) . They did 12 . � 1 2 ˆ � Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥ δ ( G ) .

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ • ˆ α ′ ( K 4 ) = 1 when properly colored. Assume ˆ δ ( G ) ≥ 4 . � 1 5 α ′ ( G ) ≥ 2 ˆ � Conj. (Wang–Li [2008]) ˆ δ ( G ) . They did 12 . � 1 2 ˆ � Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥ δ ( G ) . � 1 2 ˆ α ′ ( G ) ≥ � Thm. (Kostochka–Yancey [2012]) ˆ . δ ( G )

Large Rainbow Matchings Conj. Ryser [1967] Latin squares of odd order have transversals (distinct entries, one per row & column). Conj. (Ryser [1967]) For odd n , proper n -edge-colorings of K n,n have rainbow perfect matchings. Def. color degree ˆ d G (  ) = #colors incident to  . min color degree ˆ max color degree ˆ δ ( G ) ; Δ ( G ) . α ′ ( G ) = mx | rainbow matching | . rainbow matching # ˆ • ˆ α ′ ( K 4 ) = 1 when properly colored. Assume ˆ δ ( G ) ≥ 4 . � 1 5 α ′ ( G ) ≥ 2 ˆ � Conj. (Wang–Li [2008]) ˆ δ ( G ) . They did 12 . � 1 2 ˆ � Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥ δ ( G ) . � 1 2 ˆ α ′ ( G ) ≥ � Thm. (Kostochka–Yancey [2012]) ˆ . δ ( G ) α ′ ( G ) ≥ ˆ δ ( G ) when n ≥ 5 . 5 ( ˆ δ ( G )) 2 . With Pfender: ˆ

Results Def. An edge-colored graph is t -tolerant if its monochromatic stars all have at most t edges.

Results Def. An edge-colored graph is t -tolerant if its monochromatic stars all have at most t edges. χ ′ ( G ) < t ( t + 1 ) n ln n . Thm. If G is t -tolerant, then ˆ χ ′ ( G ) ≥ t Also, examples exist with ˆ 2 ( n − 1 ) .

Results Def. An edge-colored graph is t -tolerant if its monochromatic stars all have at most t edges. χ ′ ( G ) < t ( t + 1 ) n ln n . Thm. If G is t -tolerant, then ˆ χ ′ ( G ) ≥ t Also, examples exist with ˆ 2 ( n − 1 ) . (where k = δ ( G ) Thm. for rainbow domination + 1 ): t classical generalized γ ( G ) ≤ n − ˆ γ ( G ) ≤ n − Δ ( G ) Berge [1962] ˆ Δ ( G ) γ ( G ) ≤ 1 t 2 n Ore [1962] (no isol.) γ ( G ) ≤ ˆ t + 1 n γ ( G ) ≤ 1 + ln ( δ ( G )+ 1 ) Arnautov [1974] γ ( G ) ≤ 1 + ln k n ˆ n δ ( G )+ 1 Payan [1975] k

Results Def. An edge-colored graph is t -tolerant if its monochromatic stars all have at most t edges. χ ′ ( G ) < t ( t + 1 ) n ln n . Thm. If G is t -tolerant, then ˆ χ ′ ( G ) ≥ t Also, examples exist with ˆ 2 ( n − 1 ) . (where k = δ ( G ) Thm. for rainbow domination + 1 ): t classical generalized γ ( G ) ≤ n − ˆ γ ( G ) ≤ n − Δ ( G ) Berge [1962] ˆ Δ ( G ) γ ( G ) ≤ 1 t 2 n Ore [1962] (no isol.) γ ( G ) ≤ ˆ t + 1 n γ ( G ) ≤ 1 + ln ( δ ( G )+ 1 ) Arnautov [1974] γ ( G ) ≤ 1 + ln k n ˆ n δ ( G )+ 1 Payan [1975] k Thm. When G is t -tolerant (and no isolated vertices), t γ ( G ) = ˆ t + 1 n ⇔ each component is a t -flar e (or monochr. C 3 ( t = 2 ) or properly edge-colored C 4 ( t = 1 )).

χ ′ ( G ) large Constructions with ˆ χ ′ ( G ) ≥ t Ex. t -tolerant edge-colored G with ˆ 2 ( n − 1 ) .

χ ′ ( G ) large Constructions with ˆ χ ′ ( G ) ≥ t Ex. t -tolerant edge-colored G with ˆ 2 ( n − 1 ) . For p ∈ N , start with a proper tp -edge-coloring of K tp . Form G by identifying color classes in t -tuples.

χ ′ ( G ) large Constructions with ˆ χ ′ ( G ) ≥ t Ex. t -tolerant edge-colored G with ˆ 2 ( n − 1 ) . For p ∈ N , start with a proper tp -edge-coloring of K tp . Form G by identifying color classes in t -tuples. α ′ ( G ) ≤ p (there are only p colors). Now ˆ

χ ′ ( G ) large Constructions with ˆ χ ′ ( G ) ≥ t Ex. t -tolerant edge-colored G with ˆ 2 ( n − 1 ) . For p ∈ N , start with a proper tp -edge-coloring of K tp . Form G by identifying color classes in t -tuples. α ′ ( G ) ≤ p (there are only p colors). Now ˆ χ ′ ( G ) ≥ 1 p | E ( G ) | ≥ t 2 ( tp − 1 ) = t 2 ( n − 1 ) = t So, ˆ 2 Δ ( G ) .