Excluding Hooks and their Complements

The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\eps(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size at least $n^{\eps(H)}$. A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both $H$ and its complement $H^{\compl}$. We show that the weaker conjecture holds if $H$ is any path with a pendant edge at its third vertex; thus we give a new infinite family of graphs for which the conjecture holds.

[1]  Nicolas Bousquet,et al.  Clique versus independent set , 2013, Eur. J. Comb..

[2]  Noga Alon,et al.  Ramsey-type Theorems with Forbidden Subgraphs , 2001, Comb..

[3]  Benny Sudakov,et al.  Density theorems for bipartite graphs and related Ramsey-type results , 2007, Comb..

[4]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[5]  Nicolas Bousquet,et al.  The Erdős-Hajnal conjecture for paths and antipaths , 2015, J. Comb. Theory, Ser. B.

[6]  Daniel Lokshtanov,et al.  Independent Set in P5-Free Graphs in Polynomial Time , 2014, SODA.

[7]  Paul Erdös,et al.  Ramsey-type theorems , 1989, Discret. Appl. Math..

[8]  Maria Chudnovsky,et al.  The Erdös-Hajnal conjecture for bull-free graphs , 2008, J. Comb. Theory, Ser. B.

[9]  Benny Sudakov,et al.  Induced Ramsey-type theorems , 2007, Electron. Notes Discret. Math..

[10]  Marthe Bonamy,et al.  The Erdös-Hajnal Conjecture for Long Holes and Antiholes , 2014, SIAM J. Discret. Math..

[11]  Jacob Fox.  A Bipartite Analogue of Dilworth’s Theorem , 2006, Order.

[12]  Paul Erdös,et al.  A Ramsey-type theorem for bipartite graphs , 2000 .

[13]  Krzysztof Choromanski.  Upper Bounds for Erdös-Hajnal Coefficients of Tournaments , 2013, J. Graph Theory.

[14]  Erik Jan van Leeuwen,et al.  Independence and Efficient Domination on P6-free Graphs , 2015, SODA.

[15]  P. Erdös.  Some remarks on the theory of graphs , 1947 .

[16]  A. Hajnal.  On spanned subgraphs of graphs , 1977 .

[17]  Paul D. Seymour,et al.  Tournaments with near-linear transitive subsets , 2014, J. Comb. Theory, Ser. B.

[18]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[19]  Krzysztof Choromanski.  Excluding pairs of tournaments , 2018, J. Graph Theory.

[20]  Aurélie Lagoutte,et al.  Clique-Stable Set separation in perfect graphs with no balanced skew-partitions , 2013, Discret. Math..

[21]  András Gyárfás.  Reflections on a Problem of Erdős and Hajnal , 2013, The Mathematics of Paul Erdős II.

[22]  J. Pach,et al.  Erdős-Hajnal-type Results on Intersection Patterns of Geometric Objects , 2008 .

[23]  Maria Chudnovsky,et al.  The Erdös–Hajnal Conjecture—A Survey , 2014, J. Graph Theory.

[24]  Krzysztof Choromanski.  EH-suprema of tournaments with no nontrivial homogeneous sets , 2015, J. Comb. Theory, Ser. B.

[25]  Mika Göös,et al.  Lower Bounds for Clique vs. Independent Set , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[26]  Maria Chudnovsky,et al.  Forcing large transitive subtournaments , 2015, J. Comb. Theory, Ser. B.

[27]  Noga Alon,et al.  Crossing patterns of semi-algebraic sets , 2005, J. Comb. Theory, Ser. A.

[28]  Anne Berry,et al.  An Introduction to Clique Minimal Separator Decomposition , 2010, Algorithms.

[29]  Bruce A. Reed,et al.  A Description of Claw-Free Perfect Graphs , 1999, J. Comb. Theory, Ser. B.

[30]  Gregory A Petsko,et al.  Dominoes , 2011, Genome Biology.

[31]  Paul D. Seymour,et al.  Excluding paths and antipaths , 2015, Comb..

[32]  Krzysztof Choromanski,et al.  Coloring tournaments with forbidden substructures , 2015, ArXiv.

[33]  Vasek Chvátal,et al.  Recognizing claw-free perfect graphs , 1987, J. Comb. Theory, Ser. B.

[34]  Vojtech Rödl.  On universality of graphs with uniformly distributed edges , 1986, Discret. Math..

[35]  Regina Tyshkevich,et al.  Line Graphs of Helly Hypergraphs , 2003, SIAM J. Discret. Math..

[36]  Marthe Bonamy,et al.  The Erd\H{o}s-Hajnal Conjecture for Long Holes and Anti-holes , 2014 .

[37]  János Pach,et al.  A bipartite analogue of Dilworth's theorem for multiple partial orders , 2009, Eur. J. Comb..