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Acyclic edge coloring of graphs with large girth Guillem Perarnau Barcelona Mathematical Days, Barcelona - November 8th, 2014 McGill University, Montreal, Canada joint work with Xing Shi Cai, Bruce Reed and Adam Bene Watts. Edge colorings A

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Acyclic edge coloring of graphs with large girth

Guillem Perarnau Barcelona Mathematical Days, Barcelona - November 8th, 2014 McGill University, Montreal, Canada joint work with Xing Shi Cai, Bruce Reed and Adam Bene Watts.

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Edge colorings

A (proper) edge k-coloring is a map c : E(G) → [k] such that for every e, f ∈ E(G) with e ∩ f = ∅, c(e) = c(f ). Chromatic index: χ′(G) = min{k : G has a proper edge k-coloring}. Let ∆ be the maximum degree of G

Guillem Perarnau Acyclic edge coloring of graphs with large girth 2 / 10

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Edge colorings

A (proper) edge k-coloring is a map c : E(G) → [k] such that for every e, f ∈ E(G) with e ∩ f = ∅, c(e) = c(f ). Chromatic index: χ′(G) = min{k : G has a proper edge k-coloring}. Let ∆ be the maximum degree of G 1- χ′(G) ≥ ∆. 2- Vizing (1964): χ′(G) ≤ ∆ + 1.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 2 / 10

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Edge colorings

A (proper) edge k-coloring is a map c : E(G) → [k] such that for every e, f ∈ E(G) with e ∩ f = ∅, c(e) = c(f ). Chromatic index: χ′(G) = min{k : G has a proper edge k-coloring}. Let ∆ be the maximum degree of G 1- χ′(G) ≥ ∆. 2- Vizing (1964): χ′(G) ≤ ∆ + 1. 3- For every color i ∈ [k], c−1(i) is a matching.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 2 / 10

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Edge colorings

A (proper) edge k-coloring is a map c : E(G) → [k] such that for every e, f ∈ E(G) with e ∩ f = ∅, c(e) = c(f ). Chromatic index: χ′(G) = min{k : G has a proper edge k-coloring}. Let ∆ be the maximum degree of G 1- χ′(G) ≥ ∆. 2- Vizing (1964): χ′(G) ≤ ∆ + 1. 3- For every color i ∈ [k], c−1(i) is a matching. What happens if we look at two colors classes?

Guillem Perarnau Acyclic edge coloring of graphs with large girth 2 / 10

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Edge colorings

A (proper) edge k-coloring is a map c : E(G) → [k] such that for every e, f ∈ E(G) with e ∩ f = ∅, c(e) = c(f ). Chromatic index: χ′(G) = min{k : G has a proper edge k-coloring}. Let ∆ be the maximum degree of G 1- χ′(G) ≥ ∆. 2- Vizing (1964): χ′(G) ≤ ∆ + 1. 3- For every color i ∈ [k], c−1(i) is a matching. What happens if we look at two colors classes?

Guillem Perarnau Acyclic edge coloring of graphs with large girth 2 / 10

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Acylic edge colorings

Gr¨ unbaum (1973): An acylic edge k-coloring is a proper edge k-coloring such that for every i, j ∈ [k], c−1(i) ∪ c−1(j) is acylic. Or equivalently, every cycle contains at least 3 colors Acyclic chromatic index: a′(G) = min{k : G has an acyclic edge k-coloring}.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 3 / 10

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Acylic edge colorings

Gr¨ unbaum (1973): An acylic edge k-coloring is a proper edge k-coloring such that for every i, j ∈ [k], c−1(i) ∪ c−1(j) is acylic. Or equivalently, every cycle contains at least 3 colors Acyclic chromatic index: a′(G) = min{k : G has an acyclic edge k-coloring}.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 3 / 10

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Acylic edge colorings

Gr¨ unbaum (1973): An acylic edge k-coloring is a proper edge k-coloring such that for every i, j ∈ [k], c−1(i) ∪ c−1(j) is acylic. Or equivalently, every cycle contains at least 3 colors Acyclic chromatic index: a′(G) = min{k : G has an acyclic edge k-coloring}.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 3 / 10

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Acylic edge colorings

Gr¨ unbaum (1973): An acylic edge k-coloring is a proper edge k-coloring such that for every i, j ∈ [k], c−1(i) ∪ c−1(j) is acylic. Or equivalently, every cycle contains at least 3 colors Acyclic chromatic index: a′(G) = min{k : G has an acyclic edge k-coloring}.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 3 / 10

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Conjecture

Does Vizing’s theorem hold for the acylic chromatic index? NO

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Conjecture

Does Vizing’s theorem hold for the acylic chromatic index? NO For every n ≥ 3, a′(K2n) ≥ 2n + 1 = ∆ + 2.

Proof: Suppose a′(G) ≤ 2n = ∆ + 1. For every color i, c−1(i) ≤ n. The average size of a color class is Ei(|c−1(i)|) = n

2n = n − 1

2 . Since n ≥ 3, there exist colors i, j such that |c−1(i)|, |c−1(j)| = n. The subgraph c−1(i) ∪ c−1(j) has 2n vertices and 2n edges: it contains a cycle. Contradiction!

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Conjecture

Does Vizing’s theorem hold for the acylic chromatic index? NO For every n ≥ 3, a′(K2n) ≥ 2n + 1 = ∆ + 2.

Proof: Suppose a′(G) ≤ 2n = ∆ + 1. For every color i, c−1(i) ≤ n. The average size of a color class is Ei(|c−1(i)|) = n

2n = n − 1

2 . Since n ≥ 3, there exist colors i, j such that |c−1(i)|, |c−1(j)| = n. The subgraph c−1(i) ∪ c−1(j) has 2n vertices and 2n edges: it contains a cycle. Contradiction!

For every graph G with maximum degree ∆ a′(G) ≤ ∆ + 2 . Conjecture (Fiamˇ cik 1978; Alon, Sudakov and Zaks 2001)

Guillem Perarnau Acyclic edge coloring of graphs with large girth 4 / 10

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Previous results and the girth

For every graph G with maximum degree ∆, Alon, McDiarmid and Reed (1991): a′(G) ≤ 64∆. Molloy and Reed (1998): a′(G) ≤ 16∆. Ndreca, Procacci and Scoppola (2012): a′(G) ≤ ⌈9.62(∆ − 1)⌉. Esperet and Parreau (2013) a′(G) ≤ 4∆ − 4. Giotis, et al. (2014) a′(G) ≤ 3.732(∆ − 1) + 1.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 5 / 10

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Previous results and the girth

For every graph G with maximum degree ∆, Alon, McDiarmid and Reed (1991): a′(G) ≤ 64∆. Molloy and Reed (1998): a′(G) ≤ 16∆. Ndreca, Procacci and Scoppola (2012): a′(G) ≤ ⌈9.62(∆ − 1)⌉. Esperet and Parreau (2013) a′(G) ≤ 4∆ − 4. Giotis, et al. (2014) a′(G) ≤ 3.732(∆ − 1) + 1. If g is the girth of G, Alon, Sudakov and Zacks (2001): if g ≥ c∆ log ∆, for some large c, a′(G) ≤ ∆ + 2. Esperet and Parreau (2013): if g ≥ 220, a′(G) ≤ ⌈3.05(∆ − 1)⌉.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 5 / 10

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Previous results and the girth

For every graph G with maximum degree ∆, Alon, McDiarmid and Reed (1991): a′(G) ≤ 64∆. Molloy and Reed (1998): a′(G) ≤ 16∆. Ndreca, Procacci and Scoppola (2012): a′(G) ≤ ⌈9.62(∆ − 1)⌉. Esperet and Parreau (2013) a′(G) ≤ 4∆ − 4. Giotis, et al. (2014) a′(G) ≤ 3.732(∆ − 1) + 1. If g is the girth of G, Alon, Sudakov and Zacks (2001): if g ≥ c∆ log ∆, for some large c, a′(G) ≤ ∆ + 2. Esperet and Parreau (2013): if g ≥ 220, a′(G) ≤ ⌈3.05(∆ − 1)⌉. ∀ε > 0, ∃gε such that if g ≥ gε a′(G) ≤ (2 + ε)∆.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 5 / 10

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Our result

For every ε > 0, there exist gε and Cε > 0, such that if G has maximum degree ∆ and girth g ≥ gε , then a′(G) ≤ (1 + ε)∆ + Cε . Theorem (Cai, P., Reed and Watts, 2014+)) We can obtain the following explicit bound on the girth gε = O(ε−2). Same techniques might be able to show a′(G) ≤ ∆ + √ ∆ · Polylog(∆). The proof can be easily adapted to acyclic list edge colorings.

Guillem Perarnau Acyclic edge coloring of graphs with large girth 6 / 10

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Sketch of the proof (1 of 3)

Preprocess: Regularizing the graph For every graph G with maximum degree ∆ and girth g there exists a ∆-regular supergraph G ′ with girth at least g. Step 1: Reserving some colors For each vertex v include the colour i ∈ [(1 + ε)∆] in S(v) independently with probability p ≈ ε/3. With positive probability we have: 1- for every vertex v, |Sv| ≤ ε

3 ∆,

2- for every edge e = uv, S(e) = S(u) ∩ S(v) has size |Se| ≥ ε2

18 ∆.

3- for every vertex v and colour i, |{u ∈ N(v) : i ∈ Su}| ≤ ε

2 ∆.

We will give each edge, the initial list of colors L0(e) = [(1 + ε)∆] \ (S(u) ∪ S(v)) . (|L0(e)| ≥ (1 + ε/3)∆)

Guillem Perarnau Acyclic edge coloring of graphs with large girth 7 / 10

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