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An upper bound on the fractional chromatic number of triangle-free subcubic graphs Chun-Hung Liu Georgia Institute of Technology May 10, 2012 Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 1 / 15

Definition An ( a : b )-coloring of a graph G is a function f which maps the vertices of G into b -element subsets of some set of size a in such a way that f ( u ) is disjoint from f ( v ) for every two adjacent vertices u and v in G . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 2 / 15

Definition An ( a : b )-coloring of a graph G is a function f which maps the vertices of G into b -element subsets of some set of size a in such a way that f ( u ) is disjoint from f ( v ) for every two adjacent vertices u and v in G . Every ( a : 1)-coloring is a proper vertex a -coloring. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 2 / 15

Definition An ( a : b )-coloring of a graph G is a function f which maps the vertices of G into b -element subsets of some set of size a in such a way that f ( u ) is disjoint from f ( v ) for every two adjacent vertices u and v in G . Every ( a : 1)-coloring is a proper vertex a -coloring. The fractional chromatic number χ f ( G ) is the infimum of a / b over all pairs of positive integers a , b such that G has an ( a : b )-coloring. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 2 / 15

Definition An ( a : b )-coloring of a graph G is a function f which maps the vertices of G into b -element subsets of some set of size a in such a way that f ( u ) is disjoint from f ( v ) for every two adjacent vertices u and v in G . Every ( a : 1)-coloring is a proper vertex a -coloring. The fractional chromatic number χ f ( G ) is the infimum of a / b over all pairs of positive integers a , b such that G has an ( a : b )-coloring. So χ f ( G ) ≤ χ ( G ). Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 2 / 15

Definition An ( a : b )-coloring of a graph G is a function f which maps the vertices of G into b -element subsets of some set of size a in such a way that f ( u ) is disjoint from f ( v ) for every two adjacent vertices u and v in G . Every ( a : 1)-coloring is a proper vertex a -coloring. The fractional chromatic number χ f ( G ) is the infimum of a / b over all pairs of positive integers a , b such that G has an ( a : b )-coloring. So χ f ( G ) ≤ χ ( G ). In fact, the infimum is the minimum. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 2 / 15

A conjecture If G has an ( a : b )-coloring, then α ( G ) ≥ bn / a . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 3 / 15

A conjecture If G has an ( a : b )-coloring, then α ( G ) ≥ bn / a . Staton (1979) proved that α ( G ) ≥ 5 n / 14 for every triangle-free subcubic graph (i.e. graph of max. degree ≤ 3). Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 3 / 15

A conjecture If G has an ( a : b )-coloring, then α ( G ) ≥ bn / a . Staton (1979) proved that α ( G ) ≥ 5 n / 14 for every triangle-free subcubic graph (i.e. graph of max. degree ≤ 3). Fajtlowicz (1978) pointed out that the generalized Petersen graph P (7 , 2) has 14 vertices and no independent set of size 6. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 3 / 15

A conjecture If G has an ( a : b )-coloring, then α ( G ) ≥ bn / a . Staton (1979) proved that α ( G ) ≥ 5 n / 14 for every triangle-free subcubic graph (i.e. graph of max. degree ≤ 3). Fajtlowicz (1978) pointed out that the generalized Petersen graph P (7 , 2) has 14 vertices and no independent set of size 6. Heckman and Thomas (2001) gave a short proof of Staton’s Theorem, and they gave the following conjecture. Conjecture: The fractional chromatic number of every triangle-free subcubic graph is at most 14 / 5 . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 3 / 15

A conjecture If G has an ( a : b )-coloring, then α ( G ) ≥ bn / a . Staton (1979) proved that α ( G ) ≥ 5 n / 14 for every triangle-free subcubic graph (i.e. graph of max. degree ≤ 3). Fajtlowicz (1978) pointed out that the generalized Petersen graph P (7 , 2) has 14 vertices and no independent set of size 6. Heckman and Thomas (2001) gave a short proof of Staton’s Theorem, and they gave the following conjecture. Conjecture: The fractional chromatic number of every triangle-free subcubic graph is at most 14 / 5 . The statement that χ f ( G ) ≤ 14 / 5 is equivalent to the statement that for every weight function defined on V ( G ), there is an independent set I of G such that the sum of weights of vertices in I is at least 5 w ( G ) / 14. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 3 / 15

Known results Let G be a triangle-free subcubic graph. (Hatami and Zhu 2009) χ f ( G ) ≤ 3 − 3 64 ≈ 2 . 953, and 1 χ f ( G ) ≤ 2 . 78571 if the girth of G is at least 7. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 4 / 15

Known results Let G be a triangle-free subcubic graph. (Hatami and Zhu 2009) χ f ( G ) ≤ 3 − 3 64 ≈ 2 . 953, and 1 χ f ( G ) ≤ 2 . 78571 if the girth of G is at least 7. (Lu and Peng 2010 + ) χ f ( G ) ≤ 3 − 3 43 ≈ 2 . 930. 2 Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 4 / 15

Known results Let G be a triangle-free subcubic graph. (Hatami and Zhu 2009) χ f ( G ) ≤ 3 − 3 64 ≈ 2 . 953, and 1 χ f ( G ) ≤ 2 . 78571 if the girth of G is at least 7. (Lu and Peng 2010 + ) χ f ( G ) ≤ 3 − 3 43 ≈ 2 . 930. 2 al 2012 + ) χ f ( G ) ≤ 32 / 11 ≈ 2 . 909. (Ferguson, Kaiser, and Kr´ 3 Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 4 / 15

Known results Let G be a triangle-free subcubic graph. (Hatami and Zhu 2009) χ f ( G ) ≤ 3 − 3 64 ≈ 2 . 953, and 1 χ f ( G ) ≤ 2 . 78571 if the girth of G is at least 7. (Lu and Peng 2010 + ) χ f ( G ) ≤ 3 − 3 43 ≈ 2 . 930. 2 al 2012 + ) χ f ( G ) ≤ 32 / 11 ≈ 2 . 909. (Ferguson, Kaiser, and Kr´ 3 Theorem (L.) The fractional chromatic number of every triangle-free subcubic graph is at most 43 / 15 ≈ 2 . 867 . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 4 / 15

Larger maximum degree Let G be a K ∆ -free graph of maximum degree ∆ other than C 2 8 and C 5 ⊠ K 2 . (King, Lu, and Peng 2012) χ f ( G ) ≤ 4 − 2 67 ≈ 3 . 9701 when 1 ∆ = 4. ( χ f ( C 2 11 ) = 4 − 1 3 ≈ 3 . 6778). (King, Lu, and Peng 2012) χ f ( G ) ≤ 5 − 2 67 ≈ 4 . 9701 when 2 ∆ = 5. ( χ f ( C 7 ⊠ K 2 ) = 5 − 1 3 ≈ 4 . 6778). (Edwards and King) χ f ( G ) ≤ 6 − 153 3431 ≈ 5 . 9554 when ∆ = 6. 3 ( χ f (( C 5 ⊠ K 3 ) − 4 v ) = 6 − 1 2 = 5 . 5). (Edwards and King) χ f ( G ) ≤ 7 − 80 889 ≈ 6 . 9100 when ∆ = 7. 4 ( χ f (( C 5 ⊠ K 3 ) − 2 v ) = 7 − 1 2 = 6 . 5). (Edwards and King) χ f ( G ) ≤ 8 − 17280 152209 ≈ 7 . 8864 when ∆ = 8. 5 ( χ f ( C 5 ⊠ K 3 ) = 8 − 1 2 = 7 . 5). (Edwards and King) χ f ( G ) ≤ 9 − 17 130 ≈ 8 . 8692 when ∆ = 9. 6 (Edwards and King) χ f ( G ) ≤ 10 − 8565625 60177971 ≈ 9 . 8577 when 7 ∆ = 10. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 5 / 15

Main ideas Find a ”good” proper 3-coloring f of G such that for each 1 ≤ i < j ≤ 3, the subgraph, denoted by G ( i , j ) , of G induced by vertices of color i and j has some ”good” structures. Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 6 / 15

Main ideas Find a ”good” proper 3-coloring f of G such that for each 1 ≤ i < j ≤ 3, the subgraph, denoted by G ( i , j ) , of G induced by vertices of color i and j has some ”good” structures. Define two maps g ( i , j ) and g ( i , j ) from V ( G ) to subsets of [14] such 1 2 that adjacent vertices receive disjoint sets, and | g ( i , j ) ( v ) | + | g ( i , j ) ( v ) | 1 2 � 4 , if f ( v ) �∈ { i , j } ; = 16 − 2 deg G ( i , j ) ( v ) − 2 · 1 I ( i , j ) ( v ) , if f ( v ) ∈ { i , j } . for some independent set I ( i , j ) ⊆ { v : deg G ( i , j ) ( v ) = 3 } of G . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 6 / 15

Main ideas Define g ( i , j ) by the map from V ( G ) to subsets of [28] such that g ( i , j ) ( v ) = g ( i , j ) ( v ) ∪ ( g ( i , j ) ( v ) + 14) for every 1 ≤ i < j ≤ 3 and 1 2 vertex v . Chun-Hung Liu (Georgia Tech) An upper bound on the fractional chromatic number of triangle-free subcubic graphs May 10, 2012 7 / 15