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

ˆ χ′(G) = min{k : G has a rainbow k-edge-coloring}

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

ˆ χ′(G) = min{k : G has a rainbow k-edge-coloring}

• Def. rainbow domination: use disjoint rainbow stars

ˆ γ(G) = min{k: V(G) covered by k disjoint rainb. stars}

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

ˆ χ′(G) = min{k : G has a rainbow k-edge-coloring}

• Def. rainbow domination: use disjoint rainbow stars

ˆ γ(G) = min{k: V(G) covered by k disjoint rainb. stars} If the edge-coloring is rainbow, then ˆ χ′(G) = χ′(G). 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
• f Kn,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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree 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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

• ˆ

α′(K4) = 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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

• ˆ

α′(K4) = 1 when properly colored. Assume ˆ δ(G) ≥ 4.

• Conj. (Wang–Li [2008]) ˆ

α′(G)≥ 1

2 ˆ

δ(G)

• . They did

5 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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

• ˆ

α′(K4) = 1 when properly colored. Assume ˆ δ(G) ≥ 4.

• Conj. (Wang–Li [2008]) ˆ

α′(G)≥ 1

2 ˆ

δ(G)

• . They did

5 12.

• Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥

1

2 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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

• ˆ

α′(K4) = 1 when properly colored. Assume ˆ δ(G) ≥ 4.

• Conj. (Wang–Li [2008]) ˆ

α′(G)≥ 1

2 ˆ

δ(G)

• . They did

5 12.

• Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥

1

2 ˆ

δ(G)

• .
• Thm. (Kostochka–Yancey [2012]) ˆ

α′(G) ≥ 1

2 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
• f Kn,n have rainbow perfect matchings.
• Def. color degree ˆ

dG() = #colors incident to . min color degree ˆ δ(G); max color degree ˆ Δ(G). rainbow matching # ˆ α′(G) = mx |rainbow matching|.

• ˆ

α′(K4) = 1 when properly colored. Assume ˆ δ(G) ≥ 4.

• Conj. (Wang–Li [2008]) ˆ

α′(G)≥ 1

2 ˆ

δ(G)

• . They did

5 12.

• Thm. (LeSaulnier-Stocker-Wenger-West [2010]) ≥

1

2 ˆ

δ(G)

• .
• Thm. (Kostochka–Yancey [2012]) ˆ

α′(G) ≥ 1

2 ˆ

δ(G)

• .

With Pfender: ˆ α′(G) ≥ ˆ δ(G) when n ≥ 5.5(ˆ δ(G))2.

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.

• Thm. If G is t-tolerant, then ˆ

χ′(G) < t(t + 1)n ln n. Also, examples exist with ˆ χ′(G) ≥ t

2(n − 1).

Results

• Def. An edge-colored graph is t-tolerant if its

monochromatic stars all have at most t edges.

• Thm. If G is t-tolerant, then ˆ

χ′(G) < t(t + 1)n ln n. Also, examples exist with ˆ χ′(G) ≥ t

2(n − 1).

• Thm. for rainbow domination

(where k = δ(G)

t

+ 1): classical generalized γ(G) ≤ n − Δ(G) Berge [1962] ˆ γ(G) ≤ n − ˆ Δ(G) γ(G) ≤ 1

2n

Ore [1962] (no isol.) ˆ γ(G) ≤

t t+1n

γ(G) ≤ 1+ln(δ(G)+1)

δ(G)+1

n

Arnautov [1974] Payan [1975]

ˆ γ(G) ≤ 1+ln k

k

n

Results

• Def. An edge-colored graph is t-tolerant if its

monochromatic stars all have at most t edges.

• Thm. If G is t-tolerant, then ˆ

χ′(G) < t(t + 1)n ln n. Also, examples exist with ˆ χ′(G) ≥ t

2(n − 1).

• Thm. for rainbow domination

(where k = δ(G)

t

+ 1): classical generalized γ(G) ≤ n − Δ(G) Berge [1962] ˆ γ(G) ≤ n − ˆ Δ(G) γ(G) ≤ 1

2n

Ore [1962] (no isol.) ˆ γ(G) ≤

t t+1n

γ(G) ≤ 1+ln(δ(G)+1)

δ(G)+1

n

Arnautov [1974] Payan [1975]

ˆ γ(G) ≤ 1+ln k

k

n

• Thm. When G is t-tolerant (and no isolated vertices),

ˆ γ(G) =

t t+1n

⇔ each component is a t-flare

(or monochr. C3 (t = 2) or properly edge-colored C4 (t = 1)).

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples.

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors).

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Ex. ˆ χ′(G) > Δ(G) + 1 can occur even for a properly n-edge-colored copy of Kn,n, where n ≡ 2 mod 4.

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Ex. ˆ χ′(G) > Δ(G) + 1 can occur even for a properly n-edge-colored copy of Kn,n, where n ≡ 2 mod 4. Latin square of order n; cover by partial transversals.

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Ex. ˆ χ′(G) > Δ(G) + 1 can occur even for a properly n-edge-colored copy of Kn,n, where n ≡ 2 mod 4. Latin square of order n; cover by partial transversals. Let k = n/2. Let A and B be Latin squares of order k, using 1, . . . , k in A and k + 1, . . . , 2k in B. Let C = A B

B A

.

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Ex. ˆ χ′(G) > Δ(G) + 1 can occur even for a properly n-edge-colored copy of Kn,n, where n ≡ 2 mod 4. Latin square of order n; cover by partial transversals. Let k = n/2. Let A and B be Latin squares of order k, using 1, . . . , k in A and k + 1, . . . , 2k in B. Let C = A B

B A

. No transversal! k odd ⇒ must use ≥ ⌈k/2⌉ positions in some quadrant; others give ≤⌊k/2⌋, so ˆ α′(G) ≤n−1.

Constructions with ˆ χ′(G) large

• Ex. t-tolerant edge-colored G with ˆ

χ′(G) ≥ t

2(n − 1).

For p ∈ N, start with a proper tp-edge-coloring of Ktp. Form G by identifying color classes in t-tuples. Now ˆ α′(G) ≤ p (there are only p colors). So, ˆ χ′(G) ≥ 1

p|E(G)| ≥ t 2(tp − 1) = t 2(n − 1) = t 2Δ(G).

Ex. ˆ χ′(G) > Δ(G) + 1 can occur even for a properly n-edge-colored copy of Kn,n, where n ≡ 2 mod 4. Latin square of order n; cover by partial transversals. Let k = n/2. Let A and B be Latin squares of order k, using 1, . . . , k in A and k + 1, . . . , 2k in B. Let C = A B

B A

. No transversal! k odd ⇒ must use ≥ ⌈k/2⌉ positions in some quadrant; others give ≤⌊k/2⌋, so ˆ α′(G) ≤n−1. Thus ˆ χ′(G) ≥

n2 n−1 > n + 1 = Δ(G) + 1.

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

• Pf. If ˆ

dG() ≤

c t+1, then deleting  decreases the color

degree of up to t ˆ dG() neighbors by at most 1.

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

• Pf. If ˆ

dG() ≤

c t+1, then deleting  decreases the color

degree of up to t ˆ dG() neighbors by at most 1. Since

• V(G−) ˆ

dG−() ≥

• V(G) ˆ

dG() − (t + 1) ˆ dG() ≥ cn − c,

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

• Pf. If ˆ

dG() ≤

c t+1, then deleting  decreases the color

degree of up to t ˆ dG() neighbors by at most 1. Since

• V(G−) ˆ

dG−() ≥

• V(G) ˆ

dG() − (t + 1) ˆ dG() ≥ cn − c, deleting  does not reduce the average color degree, and G −  is t-tolerant. Iterate to reach H.

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

• Pf. If ˆ

dG() ≤

c t+1, then deleting  decreases the color

degree of up to t ˆ dG() neighbors by at most 1. Since

• V(G−) ˆ

dG−() ≥

• V(G) ˆ

dG() − (t + 1) ˆ dG() ≥ cn − c, deleting  does not reduce the average color degree, and G −  is t-tolerant. Iterate to reach H. Cor. ˆ α′(G) ≥

• m

nt(t+1)

• , where G has m edges.

Upper Bound for ˆ χ′(G) – Lemmas

• Lem. For t ∈ N and c ∈ R with c > 0, every t-tolerant

edge-colored G with average color degree ≥ c has a t-tolerant edge-colored subgraph H with ˆ δ(H) >

c t+1.

• Pf. If ˆ

dG() ≤

c t+1, then deleting  decreases the color

degree of up to t ˆ dG() neighbors by at most 1. Since

• V(G−) ˆ

dG−() ≥

• V(G) ˆ

dG() − (t + 1) ˆ dG() ≥ cn − c, deleting  does not reduce the average color degree, and G −  is t-tolerant. Iterate to reach H. Cor. ˆ α′(G) ≥

• m

nt(t+1)

• , where G has m edges.
• Pf. t-tolerant ⇒ ˆ

dG()≥ dG()/t. With degree-sum 2m, the average color degree is ≥ 2m/(nt). The lemma yields H with ˆ δ(H) >

2m nt(t+1). Now ˆ

α′(H) ≥

• m

nt(t+1)

• .

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

• Pf. We may assume G is an edge-coloring of Kn.

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

• Pf. We may assume G is an edge-coloring of Kn.

Let F0 = G and 0 = 1. For  > 0, obtain F from F−1 by deleting a large rainbow matching M−1; let  = |E(F)| (

n 2) .

By the corollary, |M−1| ≥ |E(F−1)|

nt(t+1) = −1 n−1 2t(t+1).

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

• Pf. We may assume G is an edge-coloring of Kn.

Let F0 = G and 0 = 1. For  > 0, obtain F from F−1 by deleting a large rainbow matching M−1; let  = |E(F)| (

n 2) .

By the corollary, |M−1| ≥ |E(F−1)|

nt(t+1) = −1 n−1 2t(t+1).

Let j be the least index such that j

n−1 2t(t+1) ≤ 1.

Fj is covered by |E(Fj)| single-edge rainbow matchings.

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

• Pf. We may assume G is an edge-coloring of Kn.

Let F0 = G and 0 = 1. For  > 0, obtain F from F−1 by deleting a large rainbow matching M−1; let  = |E(F)| (

n 2) .

By the corollary, |M−1| ≥ |E(F−1)|

nt(t+1) = −1 n−1 2t(t+1).

Let j be the least index such that j

n−1 2t(t+1) ≤ 1.

Fj is covered by |E(Fj)| single-edge rainbow matchings. Thus ˆ χ′(G) ≤ j + |E(Fj)|.

Upper Bound for ˆ χ′(G) – Theorem

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ χ′(G) < t(t + 1)n ln n.

• Pf. We may assume G is an edge-coloring of Kn.

Let F0 = G and 0 = 1. For  > 0, obtain F from F−1 by deleting a large rainbow matching M−1; let  = |E(F)| (

n 2) .

By the corollary, |M−1| ≥ |E(F−1)|

nt(t+1) = −1 n−1 2t(t+1).

Let j be the least index such that j

n−1 2t(t+1) ≤ 1.

Fj is covered by |E(Fj)| single-edge rainbow matchings. Thus ˆ χ′(G) ≤ j + |E(Fj)|. It remains to bound j and |E(Fj)|.

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Now 0 = 1 yields  ≤

• 1 −

1 nt(t+1)

 < e

− nt(t+1).

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Now 0 = 1 yields  ≤

• 1 −

1 nt(t+1)

 < e

− nt(t+1).

We have j ≤ 2t(t+1)

n−1

= 2t(t+1)

n−1

< j−1 < e

−j+1 nt(t+1).

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Now 0 = 1 yields  ≤

• 1 −

1 nt(t+1)

 < e

− nt(t+1).

We have j ≤ 2t(t+1)

n−1

= 2t(t+1)

n−1

< j−1 < e

−j+1 nt(t+1).

Finally, we compute j + j n

2

• <

nt(t + 1) ln

n−1 2t(t+1) + 1 + 2t(t+1) n−1 n(n−1) 2

< t(t + 1)n ln(n − 1).

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Now 0 = 1 yields  ≤

• 1 −

1 nt(t+1)

 < e

− nt(t+1).

We have j ≤ 2t(t+1)

n−1

= 2t(t+1)

n−1

< j−1 < e

−j+1 nt(t+1).

Finally, we compute j + j n

2

• <

nt(t + 1) ln

n−1 2t(t+1) + 1 + 2t(t+1) n−1 n(n−1) 2

< t(t + 1)n ln(n − 1). Thus ˆ χ′(G) < t(t + 1)n ln n.

Upper Bound for ˆ χ′(G) – Completion

Note  n

2

= |E(F−1)| − |M−1| ≤ −1 n

2

1 −

1 nt(t+1)

• .

Now 0 = 1 yields  ≤

• 1 −

1 nt(t+1)

 < e

− nt(t+1).

We have j ≤ 2t(t+1)

n−1

= 2t(t+1)

n−1

< j−1 < e

−j+1 nt(t+1).

Finally, we compute j + j n

2

• <

nt(t + 1) ln

n−1 2t(t+1) + 1 + 2t(t+1) n−1 n(n−1) 2

< t(t + 1)n ln(n − 1). Thus ˆ χ′(G) < t(t + 1)n ln n. Note: Below: a t-tolerant edge-colored graph G with avg color degree (t + 1)/2, but ˆ δ(H) ≤ 1 for all H ⊆ G.

• 1

t y1 yt

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t.

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t. Set p = ln k

k . Form A by including each vertex with

probability p, so E(|A|) = pn. Let B = V(G) −

• ∈A V(S).

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t. Set p = ln k

k . Form A by including each vertex with

probability p, so E(|A|) = pn. Let B = V(G) −

• ∈A V(S).

Note that ˆ γ(G) ≤ |A| + |B|.

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t. Set p = ln k

k . Form A by including each vertex with

probability p, so E(|A|) = pn. Let B = V(G) −

• ∈A V(S).

Note that ˆ γ(G) ≤ |A| + |B|. Note  ∈ B if  / ∈ A and [ / ∈ A or  / ∈ S for  ∈ N()]. Thus P( ∈ B) ≤ (1 − p)[(1 − p) + p(1 − 1/t)]δ(G).

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t. Set p = ln k

k . Form A by including each vertex with

probability p, so E(|A|) = pn. Let B = V(G) −

• ∈A V(S).

Note that ˆ γ(G) ≤ |A| + |B|. Note  ∈ B if  / ∈ A and [ / ∈ A or  / ∈ S for  ∈ N()]. Thus P( ∈ B) ≤ (1 − p)[(1 − p) + p(1 − 1/t)]δ(G). Now P(∈B)≤ (1−p)(1− p

t )δ(G) ≤ e−pe−δ(G)p/t = e−pk = 1 k.

Arnautov–Payan bound — Generalization

• Thm. If G is an n-vertex t-tolerant edge-colored graph,

then ˆ γ(G) ≤ 1+ln k

k

n, where k = δ(G)

t

+ 1.

• Pf. For  ∈ V(G), form S at  by including a random

incident edge of each color. Note P( ∈ E(S)) ≥ 1/t. Set p = ln k

k . Form A by including each vertex with

probability p, so E(|A|) = pn. Let B = V(G) −

• ∈A V(S).

Note that ˆ γ(G) ≤ |A| + |B|. Note  ∈ B if  / ∈ A and [ / ∈ A or  / ∈ S for  ∈ N()]. Thus P( ∈ B) ≤ (1 − p)[(1 − p) + p(1 − 1/t)]δ(G). Now P(∈B)≤ (1−p)(1− p

t )δ(G) ≤ e−pe−δ(G)p/t = e−pk = 1 k.

Thus E(|B|) ≤ n/k. We conclude E(|A ∪ B|) ≤ (1+ln k)

k

n.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

• Let U = independent set of size n − k.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

• 1

k Let U = independent set of size n − k. Let W = {1, . . . , k}, centers of monochromatic stars.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

• 1

k Let U = independent set of size n − k. Let W = {1, . . . , k}, centers of monochromatic stars. Make W a clique using edges with distinct new colors.

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

• 1

k Let U = independent set of size n − k. Let W = {1, . . . , k}, centers of monochromatic stars. Make W a clique using edges with distinct new colors. Now ˆ d() = k for all , but ˆ γ(G) = n − k. (No rainbow star covers two vertices of U.)

Berge’s bound — Generalization

Prop. ˆ γ(G) ≤ n − ˆ Δ(G), which is sharp even for highly tolerant graphs with connectivity ˆ Δ(G).

• Pf. A largest rainbow star covers ˆ

Δ(G) + 1 vertices. Sharpness: Construction with ˆ Δ(G) = k.

• 1

k Let U = independent set of size n − k. Let W = {1, . . . , k}, centers of monochromatic stars. Make W a clique using edges with distinct new colors. Now ˆ d() = k for all , but ˆ γ(G) = n − k. Note: ˆ γ(G)/n → 1, but t/n → 1.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

• Pf. (of Thm) From the family F, consider F ∈ F with

center F. A largest rainbow star in F has ˆ dF(F) edges.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

• Pf. (of Thm) From the family F, consider F ∈ F with

center F. A largest rainbow star in F has ˆ dF(F) edges. Let F′ consist of a largest rainbow star inside each member of F. Let s =

• F∈F ˆ

dF(F) and k = |F′|.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

• Pf. (of Thm) From the family F, consider F ∈ F with

center F. A largest rainbow star in F has ˆ dF(F) edges. Let F′ consist of a largest rainbow star inside each member of F. Let s =

• F∈F ˆ

dF(F) and k = |F′|. F′ covers k + s vertices with k rainbow stars. Add 1-vertex stars; now ˆ γ(G) ≤ n − s. Note that s ≥ k.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

• Pf. (of Thm) From the family F, consider F ∈ F with

center F. A largest rainbow star in F has ˆ dF(F) edges. Let F′ consist of a largest rainbow star inside each member of F. Let s =

• F∈F ˆ

dF(F) and k = |F′|. F′ covers k + s vertices with k rainbow stars. Add 1-vertex stars; now ˆ γ(G) ≤ n − s. Note that s ≥ k. If F ∈ F, then |V(F)| ≤ t · ˆ dF(F) + 1. Summing over F yields n ≤ ts + k ≤ (t + 1)s.

Ore’s Bound — Generalization

Thm. ˆ γ(G) ≤

t t+1n when G is t-tolerant and δ(G) ≥ 1.

• Lem. If G has no isolated vertices, then V(G) can be

covered by a family F of disjoint nontrivial stars in G.

• Pf. A smallest edge cover has no three edges forming a

triangle or a path, so it forms disjoint nontrival stars.

• Pf. (of Thm) From the family F, consider F ∈ F with

center F. A largest rainbow star in F has ˆ dF(F) edges. Let F′ consist of a largest rainbow star inside each member of F. Let s =

• F∈F ˆ

dF(F) and k = |F′|. F′ covers k + s vertices with k rainbow stars. Add 1-vertex stars; now ˆ γ(G) ≤ n − s. Note that s ≥ k. If F ∈ F, then |V(F)| ≤ t · ˆ dF(F) + 1. Summing over F yields n ≤ ts + k ≤ (t + 1)s. Thus ˆ γ(G) ≤ n − s ≤

t t+1n, since s ≥ 1 t+1n.

Characterization of Equality

• Def. The t-corona H ◦ t is formed by adding t pendant

edges at each vertex of H. A t-flare is an edge-colored t-corona H ◦ t that is t-tolerant and, for each vertex of H, has the same color on all t new pendant edges there.

Characterization of Equality

• Def. The t-corona H ◦ t is formed by adding t pendant

edges at each vertex of H. A t-flare is an edge-colored t-corona H ◦ t that is t-tolerant and, for each vertex of H, has the same color on all t new pendant edges there.

• No rainbow star covers two leaves, so ˆ

γ(G) =

t t+1n.

Characterization of Equality

• Def. The t-corona H ◦ t is formed by adding t pendant

edges at each vertex of H. A t-flare is an edge-colored t-corona H ◦ t that is t-tolerant and, for each vertex of H, has the same color on all t new pendant edges there.

• No rainbow star covers two leaves, so ˆ

γ(G) =

t t+1n.

• Thm. Equality

⇒ every component is a t-flare

(or monochr. C3 (t = 2) or properly edge-colored C4 (t = 1)).

Characterization of Equality

• Def. The t-corona H ◦ t is formed by adding t pendant

edges at each vertex of H. A t-flare is an edge-colored t-corona H ◦ t that is t-tolerant and, for each vertex of H, has the same color on all t new pendant edges there.

• No rainbow star covers two leaves, so ˆ

γ(G) =

t t+1n.

• Thm. Equality

⇒ every component is a t-flare

(or monochr. C3 (t = 2) or properly edge-colored C4 (t = 1)).

• For t = 1 (where ˆ

γ(G) = γ(G)), Payan–Xuong [1982] and Fink–Jacobson–Kinch–Roberts [1985] char’zd γ(G) = n/2.

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •
• Reduce to connected G; let T be any spanning tree.

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •
• Reduce to connected G; let T be any spanning tree.

Let  be a nonleaf vertex in T. Can  have no leaf nbr?

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •
• Reduce to connected G; let T be any spanning tree.

Let  be a nonleaf vertex in T. Can  have no leaf nbr?

• C3

C1 C2 

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •
• Reduce to connected G; let T be any spanning tree.

Let  be a nonleaf vertex in T. Can  have no leaf nbr?

• C3

C1 C2  If (t+1) ∤ |V(C)|, then strict inequality for C (and G).

Sketch of Characterizing Equality in ˆ γ(G) ≤

t t+1n

• •
• •
• Reduce to connected G; let T be any spanning tree.

Let  be a nonleaf vertex in T. Can  have no leaf nbr?

• C3

C1 C2  If (t+1) ∤ |V(C)|, then strict inequality for C (and G). Now (t+1) ∤ n, and again the inequality is strict for G.

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2 

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves.

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Simplifies to ℓ+1

t+1 ≥ k.

Also t-tolerant ⇒ k ≥ ℓ

t.

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Simplifies to ℓ+1

t+1 ≥ k.

Also t-tolerant ⇒ k ≥ ℓ

t.

From ℓ+1

t+1 ≥ k ≥ ℓ t, conclude ℓ = t and k = 1.

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Simplifies to ℓ+1

t+1 ≥ k.

Also t-tolerant ⇒ k ≥ ℓ

t.

From ℓ+1

t+1 ≥ k ≥ ℓ t, conclude ℓ = t and k = 1.

∴ Every spanning tree is a t-flare.

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Simplifies to ℓ+1

t+1 ≥ k.

Also t-tolerant ⇒ k ≥ ℓ

t.

From ℓ+1

t+1 ≥ k ≥ ℓ t, conclude ℓ = t and k = 1.

∴ Every spanning tree is a t-flare. Claim: No other edges at leaves of a spanning tree T. (Otherwise, some spanning tree is not a t-flare.)

Idea, continued

∴  has leaf nbr(s), say ℓ of them, with k colors.

• C1

C2  Now T has a rainbow star F at  with k leaves. ˆ γ(G) ≤ 1 + ℓ − k +

• ˆ

γ(C)

t t+1n ≤ 1 + ℓ − k + t t+1(n − ℓ − 1)

Simplifies to ℓ+1

t+1 ≥ k.

Also t-tolerant ⇒ k ≥ ℓ

t.

From ℓ+1

t+1 ≥ k ≥ ℓ t, conclude ℓ = t and k = 1.

∴ Every spanning tree is a t-flare. Claim: No other edges at leaves of a spanning tree T. (Otherwise, some spanning tree is not a t-flare.)

(The exceptions: monochr. C3 and properly colored C4.)

Open Problems

• Improve the bounds on the maximum value of the

rainbow edge-chromatic number ˆ χ′(G) among t-tolerant n-vertex graphs.

Open Problems

• Improve the bounds on the maximum value of the

rainbow edge-chromatic number ˆ χ′(G) among t-tolerant n-vertex graphs.

• Generalize other bounds on the domination number

γ(G) to the rainbow domination number ˆ γ(G).

Open Problems

• Improve the bounds on the maximum value of the

rainbow edge-chromatic number ˆ χ′(G) among t-tolerant n-vertex graphs.

• Generalize other bounds on the domination number

γ(G) to the rainbow domination number ˆ γ(G).

• Generalize other problems on ordinary graphs to the

setting of edge-colored graphs. (T urán problems, Ramsey problems, etc.)

Open Problems

• Improve the bounds on the maximum value of the

rainbow edge-chromatic number ˆ χ′(G) among t-tolerant n-vertex graphs.

• Generalize other bounds on the domination number

γ(G) to the rainbow domination number ˆ γ(G).

• Generalize other problems on ordinary graphs to the

setting of edge-colored graphs. (T urán problems, Ramsey problems, etc.)

Reference:

• T. D. LeSaulnier and D. B. West,

Rainbow edge-coloring and rainbow domination, Discrete Math. (2012), DOI: 10.1016/j.disc.2012.03.014.