11/4-colorability of subcubic triangle-free graphs Zden ek Dvo r - PowerPoint PPT Presentation

11/4-colorability of subcubic triangle-free graphs Zdenˇ ek Dvoˇ r´ ak Bernard Lidick´ y Luke Postle AMS October 10, 2020

Motivation Theorem (4CT - Appel, Haken (1976)) Every planar graph is 4 -colorable. Also Robertson, Sanders, Seymour, Thomas (1997) 2

Motivation Theorem (4CT - Appel, Haken (1976)) Every planar graph is 4 -colorable. Also Robertson, Sanders, Seymour, Thomas (1997) 4CT implies α ( G ) ≥ n / 4 Show without 4CT α ( G ) ≥ n / 4 (Erd˝ os-Vizing conj.) 2

Motivation Theorem (4CT - Appel, Haken (1976)) 2 2 Every planar graph is 4 -colorable. 2 2 2 Also Robertson, Sanders, Seymour, Thomas (1997) 3 4 4 3 1 4CT implies α ( G ) ≥ n / 4 2 2 Show without 4CT α ( G ) ≥ n / 4 (Erd˝ os-Vizing conj.) Assign weights to V ( G ). 4CT implies independent set with ≥ 1 / 4 of the weight. Is it true without 4CT? 2

Motivation Theorem (4CT - Appel, Haken (1976)) 1 1 Every planar graph is 4 -colorable. 1 1 1 Also Robertson, Sanders, Seymour, Thomas (1997) 1 1 1 1 1 4CT implies α ( G ) ≥ n / 4 1 1 Show without 4CT α ( G ) ≥ n / 4 (Erd˝ os-Vizing conj.) Assign weights to V ( G ). 4CT implies independent set with ≥ 1 / 4 of the weight. Is it true without 4CT? 2

Motivation Theorem (4CT - Appel, Haken (1976)) 1 1 Every planar graph is 4 -colorable. 1 1 1 Also Robertson, Sanders, Seymour, Thomas (1997) 1 1 1 1 1 4CT implies α ( G ) ≥ n / 4 1 1 Show without 4CT α ( G ) ≥ n / 4 (Erd˝ os-Vizing conj.) Assign weights to V ( G ). 4CT implies independent set with ≥ 1 / 4 of the weight. Is it true without 4CT? Without 4CT α ( G ) ≥ 3 n 13 ≈ 0 . 23076 (Cranston and Rabern (2016)) 2

Motivation Theorem (4CT - Appel, Haken (1976)) 2 2 Every planar graph is 4 -colorable. 2 2 2 Also Robertson, Sanders, Seymour, Thomas (1997) 3 4 4 3 1 4CT implies α ( G ) ≥ n / 4 2 2 Show without 4CT α ( G ) ≥ n / 4 (Erd˝ os-Vizing conj.) Assign weights to V ( G ). 4CT implies independent set with ≥ 1 / 4 of the weight. Is it true without 4CT? Without 4CT α ( G ) ≥ 3 n 13 ≈ 0 . 23076 (Cranston and Rabern (2016)) Weighted independent set leads to fractional coloring. 2

Fractional coloring using linear programming I ( G ) are all independent sets v 5 coloring v 1 v 4  minimize � I ∈I ( G ) x ( I )   P � ∀ v subject to I ∋ v x ( I ) = 1  x ∈ { 0 , 1 } I ( G )  v 2 v 3 |I ( C 5 ) | = 11 3

Fractional coloring using linear programming I ( G ) are all independent sets v 5 coloring v 1 v 4  minimize � I ∈I ( G ) x ( I )   P � ∀ v subject to I ∋ v x ( I ) = 1  x ∈ { 0 , 1 } I ( G )  v 2 v 3 |I ( C 5 ) | = 11 • x (1 , 3) = x (2 , 4) = x (5) = 1 χ ( G ) = 3 3

Fractional coloring using linear programming I ( G ) are all independent sets v 5 fractional coloring v 1 v 4  minimize � I ∈I ( G ) x ( I )   LP � ∀ v subject to I ∋ v x ( I ) = 1  x ∈ { 0 , 1 } I ( G )  x ∈ [0 , 1] I ( G ) v 2 v 3 |I ( C 5 ) | = 11 • x (1 , 3) = x (2 , 4) = x (5) = 1 χ ( G ) = 3 3

Fractional coloring using linear programming I ( G ) are all independent sets v 5 fractional coloring v 1 v 4  minimize � I ∈I ( G ) x ( I )   LP � ∀ v subject to I ∋ v x ( I ) = 1  x ∈ { 0 , 1 } I ( G )  x ∈ [0 , 1] I ( G ) v 2 v 3 |I ( C 5 ) | = 11 • x (1 , 3) = x (2 , 4) = x (5) = 1 χ ( G ) = 3 • x (1 , 3) = x (2 , 4) = x (3 , 5) = x (1 , 4) = x (2 , 5) = 1 / 2 χ f ( G ) = 2 . 5 3

Fractional coloring using linear programming I ( G ) are all independent sets v 5 fractional coloring v 1 v 4  minimize � I ∈I ( G ) x ( I )   LP � ∀ v subject to I ∋ v x ( I ) = 1  x ∈ { 0 , 1 } I ( G )  x ∈ [0 , 1] I ( G ) v 2 v 3 |I ( C 5 ) | = 11 • x (1 , 3) = x (2 , 4) = x (5) = 1 χ ( G ) = 3 • x (1 , 3) = x (2 , 4) = x (3 , 5) = x (1 , 4) = x (2 , 5) = 1 / 2 χ f ( G ) = 2 . 5 • χ ( G ) ≥ χ f ( G ) ≥ | V ( G ) | /α ( G ) 3

Equivalent definitions for fractional coloring v 5 G is fractionally k -colorable if exists ϕ • ϕ ( v ) ⊂ [0 , k ) with | ϕ ( v ) | = 1 v 1 v 4 χ f ( G ) = 5 2 and ϕ ( u ) ∩ ϕ ( v ) = ∅ for uv ∈ E ( G ) v 2 v 3 1 5 0 2 4

Equivalent definitions for fractional coloring v 5 G is fractionally k -colorable if exists ϕ • ϕ ( v ) ⊂ [0 , k ) with | ϕ ( v ) | = 1 v 1 v 4 • ∃ a b = k , ϕ ( v ) ⊂ { 1 , . . . , a } , | ϕ ( v ) | = b χ f ( G ) = 5 2 and ϕ ( u ) ∩ ϕ ( v ) = ∅ for uv ∈ E ( G ) v 2 v 3 1 2 3 4 5 4

Equivalent definitions for fractional coloring v 5 G is fractionally k -colorable if exists ϕ • ϕ ( v ) ⊂ [0 , k ) with | ϕ ( v ) | = 1 v 1 v 4 • ∃ a b = k , ϕ ( v ) ⊂ { 1 , . . . , a } , | ϕ ( v ) | = b χ f ( G ) = 5 • ϕ ( v ) ⊂ [0 , a ) with | ϕ ( v ) | = b , a 2 b = k and ϕ ( u ) ∩ ϕ ( v ) = ∅ for uv ∈ E ( G ) v 2 v 3 2 0 5 4

Equivalent definitions for fractional coloring v 5 G is fractionally k -colorable if exists ϕ • ϕ ( v ) ⊂ [0 , k ) with | ϕ ( v ) | = 1 v 1 v 4 • ∃ a b = k , ϕ ( v ) ⊂ { 1 , . . . , a } , | ϕ ( v ) | = b χ f ( G ) = 5 • ϕ ( v ) ⊂ [0 , a ) with | ϕ ( v ) | = b , a 2 b = k and ϕ ( u ) ∩ ϕ ( v ) = ∅ for uv ∈ E ( G ) v 2 v 3 Theorem (Hilton, Rado, Scott (1973)) χ f ( G ) < 5 for any planar G. (But no c < 5 with χ f ( G ) < c for all planar graphs G.) 2 0 5 Theorem (Cranston and Rabern (2017)) Planar graphs are 9 2 -colorable. (Without using 4CT.) 4

Planar triangle-free graphs Theorem (Gr¨ otzsch (1959)) Every triangle-free planar graph G is 3 -colorable. α ( G ) ≥ n / 3 5

Planar triangle-free graphs Theorem (Gr¨ otzsch (1959)) Every triangle-free planar graph G is 3 -colorable. α ( G ) ≥ n / 3 α ( G ) ≥ n / 3 + 1 Steinberg and Tovey (1993) ∃ G : α ( G ) ≤ n / 3 + 1 (and ∆( G ) = 4) Jones (1990) 5

Planar triangle-free graphs Theorem (Gr¨ otzsch (1959)) Every triangle-free planar graph G is 3 -colorable. α ( G ) ≥ n / 3 α ( G ) ≥ n / 3 + 1 Steinberg and Tovey (1993) α = 9 = 3 8 · 24 ∃ G : α ( G ) ≤ n / 3 + 1 (and ∆( G ) = 4) Jones (1990) Question (Albertson, Bollob´ as, Tucker (1976)) � 1 3 , 3 � Find s ∈ s.t. every subcubic triangle-free planar graph G has α ( G ) ≥ sn? 8 5

Planar triangle-free graphs Theorem (Gr¨ otzsch (1959)) Every triangle-free planar graph G is 3 -colorable. α ( G ) ≥ n / 3 α ( G ) ≥ n / 3 + 1 Steinberg and Tovey (1993) α = 9 = 3 8 · 24 ∃ G : α ( G ) ≤ n / 3 + 1 (and ∆( G ) = 4) Jones (1990) Question (Albertson, Bollob´ as, Tucker (1976)) � 1 3 , 3 � Find s ∈ s.t. every subcubic triangle-free planar graph G has α ( G ) ≥ sn? 8 • s = 5 14 ≈ 0 . 35714 Staton (1979) No planarity condition! • s = 3 8 = 0 . 375 Heckman and Thomas (2006) 5

Subcubic triangle-free graphs If G is a subcubic triangle-free graph α = 5 n • α ( G ) ≥ 5 n 14 ≈ 0 . 35714 n Staton (1979) 14 • α ( G ) ≥ 11 n − 4 ≈ 0 . 3666 n Fraughaugh and Locke (1995) 30 • α ( G ) ≥ 3 n 8 = 0 . 375 n Cames van Batenburg, Goedgebeur, Joret (2020) if G is avoids 6 exceptional graphs. All non-planar, containing 5-cycles. (Infinitely many 3-connected tight examples.) 6

From α to χ f If G is fractionally 1 s -colorable, it has α ( G ) ≥ sn . α ( C 5 ) = 2 χ f ( C 5 ) = 5 5 n 2 If α ( G ) = sn , is G fractionally 1 s -colorable? Conjecture (Heckman and Thomas (2001)) Every subcubic triangle-free graph is fractionally 14 / 5 -colorable. Conjecture (Heckman and Thomas (2006)) Every subcubic triangle-free planar graph is fractionally 8 / 3 -colorable. 7

Conjecture (Heckman and Thomas (2001)) Every subcubic triangle-free graph is fractionally 14 / 5 -colorable. χ f ( F (1) 14 ) = χ f ( F (1) 14 ) = 14 / 5 F (1) F (2) 14 14 8

Conjecture (Heckman and Thomas (2001)) Every subcubic triangle-free graph is fractionally 14 / 5 -colorable. χ f ( F (1) 14 ) = χ f ( F (1) 14 ) = 14 / 5 • 3 − 3 64 ≈ 2 . 953 Hatami, Zhu (2009) • 3 − 3 43 ≈ 2 . 930 Lu, Peng (2012) 32 • 11 ≈ 2 . 909 Furgeson, Kaiser, Kr´ al’ (2014) 43 • 15 ≈ 2 . 867 Chun-Hung Liu (2014) 14 • 5 = 2 . 8 Dvoˇ r´ ak, Sereni, Volec (2014) F (1) F (2) 14 14 8

Conjecture (Heckman and Thomas (2001)) Every subcubic triangle-free graph is fractionally 14 / 5 -colorable. χ f ( F (1) 14 ) = χ f ( F (1) 14 ) = 14 / 5 • 3 − 3 64 ≈ 2 . 953 Hatami, Zhu (2009) • 3 − 3 43 ≈ 2 . 930 Lu, Peng (2012) 32 • 11 ≈ 2 . 909 Furgeson, Kaiser, Kr´ al’ (2014) 43 • 15 ≈ 2 . 867 Chun-Hung Liu (2014) 14 • 5 = 2 . 8 Dvoˇ r´ ak, Sereni, Volec (2014) F (1) F (2) 14 14 11 • 4 = 2 . 75 Dvoˇ r´ ak, L., Postle, if not → Theorem (Dvoˇ r´ ak, L., Postle (2020+)) Every subcubic triangle-free graph avoiding F (1) 14 and F (1) 14 is fractionally 11 / 4 -colorable. 8

Theorem (Dvoˇ r´ ak, L., Postle (2020+)) Every subcubic triangle-free graph avoiding F (1) 14 and F (1) 14 is fractionally 11 / 4 -colorable. 3 1 2 2 1 3 F (1) F (2) F 11 F 22 14 14 χ f ( F (1) 14 ) = χ f ( F (2) 14 ) = 14 / 5 χ f ( F 22 ) = χ f ( F 11 ) = 11 / 4 9