Lucas Pastor November 15 2017 Joint-work with Rmi de Joannis de - PowerPoint PPT Presentation

Coloring squares of claw-free graphs Lucas Pastor November 15 2017 Joint-work with Rémi de Joannis de Verclos and Ross J. Kang 1

A (proper) k -coloring of G is an assignement of colors { 1 , . . . , k } to the vertices of G such that any two adjacent vertices receive a different color. 2

A (proper) k -coloring of G is an assignement of colors { 1 , . . . , k } to the vertices of G such that any two adjacent vertices receive a different color. 2

A (proper) k -coloring of G is an assignement of colors { 1 , . . . , k } to the vertices of G such that any two adjacent vertices receive a different color. 1 2 1 3 2

A (proper) k -coloring of G is an assignement of colors { 1 , . . . , k } to the vertices of G such that any two adjacent vertices receive a different color. 1 2 1 3 The chromatic number , χ ( G ), is the smallest k such that G is k -colorable. 2

A (proper) k -edge-coloring of G is an assignment of colors { 1 , . . . , k } to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color. 3

A (proper) k -edge-coloring of G is an assignment of colors { 1 , . . . , k } to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color. 3

A (proper) k -edge-coloring of G is an assignment of colors { 1 , . . . , k } to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color. 2 1 3 2 3

A (proper) k -edge-coloring of G is an assignment of colors { 1 , . . . , k } to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color. 2 1 3 2 The chromatic index , χ ′ ( G ), is the smallest k such that G is k -edge-colorable. 3

Note that in an edge coloring, each color class is a matching . 4

Note that in an edge coloring, each color class is a matching . 1 4

Note that in an edge coloring, each color class is a matching . 3 4

Note that in an edge coloring, each color class is a matching . 2 2 4

Note that in an edge coloring, each color class is a matching . 2 2 But not necessarily an induced matching ! 4

A strong k -edge-coloring of G is a k -edge-coloring where each color class is an induced matching. 5

A strong k -edge-coloring of G is a k -edge-coloring where each color class is an induced matching. 5

A strong k -edge-coloring of G is a k -edge-coloring where each color class is an induced matching. 2 1 3 4 5

A strong k -edge-coloring of G is a k -edge-coloring where each color class is an induced matching. 2 1 3 4 The strong chromatic index , χ ′ s ( G ), is the smallest k such that G is strong k -edge-colorable. 5

Questions Given a graph G with maximum degree ∆( G ). 6

Questions Given a graph G with maximum degree ∆( G ). χ ′ s ( G ) 6

Questions Given a graph G with maximum degree ∆( G ). χ ′ s ( G ) ≤ upper bound 6

Questions Given a graph G with maximum degree ∆( G ). lower bound ≤ χ ′ s ( G ) ≤ upper bound 6

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. e 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. e 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. ∆ e ∆ 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. ∆ ∆ − 1 e ∆ 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. ∆ ∆ ∆ − 1 ∆ − 1 e ∆ ∆ 7

Upper bound Pick any edge e , and look at how large can be its neighborhood at distance 2. ∆ ∆ ∆ − 1 ∆ − 1 e ∆ ∆ s ( G ) ≤ 2∆(∆ − 1) + 1 = 2∆ 2 − 2∆ + 1 . χ ′ 7

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ ∆ 2 ∆ ∆ 2 2 ∆ ∆ 2 2 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ ∆ 2 ∆ ∆ 2 2 ∆ ∆ 2 2 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ ∆ 2 ∆ ∆ 2 2 ∆ ∆ 2 2 In this graph, any pair of edges is at distance at most 2. There are 4 ∆ 2 edges in G . 5 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ ∆ 2 1 4 ∆2 ∆ ∆ 2 2 ∆ ∆ 2 2 In this graph, any pair of edges is at distance at most 2. There are 4 ∆ 2 edges in G . 5 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: s ( G ) = 5 4∆ 2 . χ ′ ∆ 2 4 ∆2 1 4 ∆2 1 ∆ ∆ 2 2 1 4 ∆2 1 4 ∆2 ∆ ∆ 2 2 1 4 ∆2 In this graph, any pair of edges is at distance at most 2. There are 4 ∆ 2 edges in G . 5 8

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: s ( G ) ≤ 5 4 ∆( G ) 2 . For any graph G , χ ′ 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: s ( G ) ≤ 5 4 ∆( G ) 2 . For any graph G , χ ′ We have an upper bound of 2∆( G ) 2 . Can we do better? 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: s ( G ) ≤ 5 4 ∆( G ) 2 . For any graph G , χ ′ We have an upper bound of 2∆( G ) 2 . Can we do better? Theorem [Molloy, Reed 1997] s ( G ) ≤ (2 − ǫ )∆( G ) 2 χ ′ 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: s ( G ) ≤ 5 4 ∆( G ) 2 . For any graph G , χ ′ We have an upper bound of 2∆( G ) 2 . Can we do better? Theorem [Molloy, Reed 1997] s ( G ) ≤ (2 − ǫ )∆( G ) 2 χ ′ for some constant ǫ = 0 . 002. 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: s ( G ) ≤ 5 4 ∆( G ) 2 . For any graph G , χ ′ We have an upper bound of 2∆( G ) 2 . Can we do better? Theorem [Molloy, Reed 1997] s ( G ) ≤ (2 − ǫ )∆( G ) 2 χ ′ for some constant ǫ = 0 . 002. The constant has been improved by Bruhn and Joos in 2015 to ǫ = 0 . 07. 9

Line-graph Given a graph G , the line-graph of G , denoted by L ( G ), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G . 10

Line-graph Given a graph G , the line-graph of G , denoted by L ( G ), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G . e 6 e 6 e 5 e 1 e 5 e 1 e 4 e 2 e 2 e 4 e 3 e 3 G L ( G ) 10

Square graph Given a graph G , the square of G , denoted by G 2 , is the graph obtained from G by adding edges between every pair of vertices at distance 2. 11

Square graph Given a graph G , the square of G , denoted by G 2 , is the graph obtained from G by adding edges between every pair of vertices at distance 2. G 11

Square graph Given a graph G , the square of G , denoted by G 2 , is the graph obtained from G by adding edges between every pair of vertices at distance 2. G G 2 11

Square graph Given a graph G , the square of G , denoted by G 2 , is the graph obtained from G by adding edges between every pair of vertices at distance 2. G G 2 11

Strong coloring • Coloring the edges of G is equivalent to coloring the vertices of L ( G ). 12

Strong coloring • Coloring the edges of G is equivalent to coloring the vertices of L ( G ). • The strong coloring of G is equivalent to color G 2 . 12

Strong coloring • Coloring the edges of G is equivalent to coloring the vertices of L ( G ). • The strong coloring of G is equivalent to color G 2 . • Hence, the strong edge coloring of G is equivalent to color the vertices of L ( G ) 2 . 12

Strong coloring • Coloring the edges of G is equivalent to coloring the vertices of L ( G ). • The strong coloring of G is equivalent to color G 2 . • Hence, the strong edge coloring of G is equivalent to color the vertices of L ( G ) 2 . Molloy and Reed’s theorem Let G be the line-graph of any simple graph, then: χ ( G 2 ) ≤ (2 − ǫ ) ω ( G ) 2 . 12

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. 13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. 13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. 13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. 13