Acyclic edge coloring of graphs with large girth Guillem Perarnau - PowerPoint PPT Presentation

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 (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

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

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

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

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

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

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

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

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

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

Conjecture Does Vizing’s theorem hold for the acylic chromatic index? NO For every n ≥ 3, a ′ ( K 2 n ) ≥ 2 n + 1 = ∆ + 2. Proof: Suppose a ′ ( G ) ≤ 2 n = ∆ + 1. For every color i , c − 1 ( i ) ≤ n . The average size of a color class is � n � 2 n = n − 1 E i ( | c − 1 ( i ) | ) = 2 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 2 n vertices and 2 n edges: it contains a cycle. Contradiction!

Conjecture Does Vizing’s theorem hold for the acylic chromatic index? NO For every n ≥ 3, a ′ ( K 2 n ) ≥ 2 n + 1 = ∆ + 2. Proof: Suppose a ′ ( G ) ≤ 2 n = ∆ + 1. For every color i , c − 1 ( i ) ≤ n . The average size of a color class is � n � 2 n = n − 1 E i ( | c − 1 ( i ) | ) = 2 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 2 n vertices and 2 n edges: it contains a cycle. Contradiction! Conjecture (Fiamˇ cik 1978; Alon, Sudakov and Zaks 2001) For every graph G with maximum degree ∆ a ′ ( G ) ≤ ∆ + 2 . Guillem Perarnau Acyclic edge coloring of graphs with large girth 4 / 10

Previous results and the girth For every graph G with maximum degree ∆, a ′ ( G ) ≤ 64∆ . Alon, McDiarmid and Reed (1991): a ′ ( G ) ≤ 16∆ . Molloy and Reed (1998): a ′ ( G ) ≤ ⌈ 9 . 62(∆ − 1) ⌉ . Ndreca, Procacci and Scoppola (2012): a ′ ( G ) ≤ 4∆ − 4 . Esperet and Parreau (2013) Giotis, et al. (2014) a ′ ( G ) ≤ 3 . 732(∆ − 1) + 1 . Guillem Perarnau Acyclic edge coloring of graphs with large girth 5 / 10

Previous results and the girth For every graph G with maximum degree ∆, a ′ ( G ) ≤ 64∆ . Alon, McDiarmid and Reed (1991): a ′ ( G ) ≤ 16∆ . Molloy and Reed (1998): a ′ ( G ) ≤ ⌈ 9 . 62(∆ − 1) ⌉ . Ndreca, Procacci and Scoppola (2012): a ′ ( G ) ≤ 4∆ − 4 . Esperet and Parreau (2013) 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

Previous results and the girth For every graph G with maximum degree ∆, a ′ ( G ) ≤ 64∆ . Alon, McDiarmid and Reed (1991): a ′ ( G ) ≤ 16∆ . Molloy and Reed (1998): a ′ ( G ) ≤ ⌈ 9 . 62(∆ − 1) ⌉ . Ndreca, Procacci and Scoppola (2012): a ′ ( G ) ≤ 4∆ − 4 . Esperet and Parreau (2013) 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) ⌉ . a ′ ( G ) ≤ (2 + ε )∆ . ∀ ε > 0 , ∃ g ε such that if g ≥ g ε Guillem Perarnau Acyclic edge coloring of graphs with large girth 5 / 10

Our result Theorem (Cai, P., Reed and Watts, 2014+)) For every ε > 0, there exist g ε and C ε > 0, such that if G has maximum degree ∆ and girth g ≥ g ε , then a ′ ( G ) ≤ (1 + ε )∆ + C ε . We can obtain the following explicit bound on the girth g ε = 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

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 , | S v | ≤ ε 3 ∆, 2- for every edge e = uv , S ( e ) = S ( u ) ∩ S ( v ) has size | S e | ≥ ε 2 18 ∆. 3- for every vertex v and colour i , |{ u ∈ N ( v ) : i ∈ S u }| ≤ ε 2 ∆. We will give each edge, the initial list of colors L 0 ( e ) = [(1 + ε )∆] \ ( S ( u ) ∪ S ( v )) . ( | L 0 ( e ) | ≥ (1 + ε/ 3)∆) Guillem Perarnau Acyclic edge coloring of graphs with large girth 7 / 10

Sketch of the proof (2 of 3) Step 2: Iterative “list” coloring Consider the lists L 0 ( e ) = [(1 + ε )∆] \ ( S ( u ) ∪ S ( v )) . Guillem Perarnau Acyclic edge coloring of graphs with large girth 8 / 10

Sketch of the proof (2 of 3) Step 2: Iterative “list” coloring After i steps we have a partial coloring Guillem Perarnau Acyclic edge coloring of graphs with large girth 8 / 10

Sketch of the proof (2 of 3) Step 2: Iterative “list” coloring These edges have a fixed color. Guillem Perarnau Acyclic edge coloring of graphs with large girth 8 / 10

Sketch of the proof (2 of 3) Step 2: Iterative “list” coloring Randomly color the uncolored edges Guillem Perarnau Acyclic edge coloring of graphs with large girth 8 / 10

Sketch of the proof (2 of 3) Step 2: Iterative “list” coloring If two incident edges have the same color . . . Guillem Perarnau Acyclic edge coloring of graphs with large girth 8 / 10