Degree Ramsey numbers of trees Bill Kinnersley Department of - PowerPoint PPT Presentation

Degree Ramsey numbers of trees Bill Kinnersley Department of Mathematics University of Illinois at Urbana-Champaign wkinner2@illinois.edu Joint work with Kevin G. Milans, Douglas B. West

Preliminaries Graph Ramsey theory: find the smallest n for which every s -edge-coloring of K n has a monochromatic G .

Preliminaries Graph Ramsey theory: find the smallest n for which every s -edge-coloring of K n has a monochromatic G . Definition s We write H → G if every s -edge-coloring of H yields a monochromatic G . (Read: H s -arrows G or H forces G .) s R ( G ; s ) = min { n : K n → G } . This is “wasteful”; if G is sparse, we don’t really need K n .

Preliminaries Graph Ramsey theory: find the smallest n for which every s -edge-coloring of K n has a monochromatic G . Definition s We write H → G if every s -edge-coloring of H yields a monochromatic G . (Read: H s -arrows G or H forces G .) s R ( G ; s ) = min { n : K n → G } . This is “wasteful”; if G is sparse, we don’t really need K n . Definition The s -color degree Ramsey number of a graph G , denoted R ∆ ( G ; s ) , is s min { ∆( H ) : H → G } .

Stars Theorem (Burr-Erd˝ os-Lovàsz, 1976) � 2 k − 2 , k even R ∆ ( K 1 , k ; 2 ) = . 2 k − 1 , k odd

Stars Theorem (Burr-Erd˝ os-Lovàsz, 1976) � 2 k − 2 , k even R ∆ ( K 1 , k ; 2 ) = . 2 k − 1 , k odd Theorem � s ( k − 1 ) , k even R ∆ ( K 1 , k ; s ) = . s ( k − 1 ) + 1 , k odd Monotonicity: if H is a subgraph of G , then R ∆ ( H ; s ) ≤ R ∆ ( G ; s ) . Hence if ∆( G ) = k , then R ∆ ( G ; s ) ≥ R ∆ ( K 1 , k ; s ) .

Stars Theorem � s ( k − 1 ) , k even R ∆ ( K 1 , k ; s ) = . s ( k − 1 ) + 1 k odd Proof (upper bound). s K 1 , s ( k − 1 )+ 1 → K 1 , k .

Stars Theorem � s ( k − 1 ) , k even R ∆ ( K 1 , k ; s ) = . s ( k − 1 ) + 1 k odd Proof (upper bound). s K 1 , s ( k − 1 )+ 1 → K 1 , k . We can do better when k is even. Let H be s ( k − 1 ) -regular and have no s ( k − 1 ) -factor [Bollobás-Saito-Wormald]. Now H → K 1 , k , since avoiding K 1 , k means finding a ( k − 1 ) -factorization.

Stars Theorem � s ( k − 1 ) , k even R ∆ ( K 1 , k ; s ) = . s ( k − 1 ) + 1 k odd Proof (lower bound). Suppose ∆( H ) < s ( k − 1 ) . By Vizing’s Theorem, H is s ( k − 1 ) -edge-colorable, so E ( H ) decomposes into s ( k − 1 ) matchings. Let each color class be the union of k − 1 matchings; this avoids K 1 , k .

Stars Theorem � s ( k − 1 ) , k even R ∆ ( K 1 , k ; s ) = . s ( k − 1 ) + 1 k odd Proof (lower bound). Suppose ∆( H ) < s ( k − 1 ) . By Vizing’s Theorem, H is s ( k − 1 ) -edge-colorable, so E ( H ) decomposes into s ( k − 1 ) matchings. Let each color class be the union of k − 1 matchings; this avoids K 1 , k . We can do better when k is odd. Suppose ∆( H ) is s ( k − 1 ) -regular. By Petersen’s Theorem, H decomposes into 2-factors. Let each color class be the union of k − 1 of these. � 2

Double-stars, two colors Definition The double-star S a , b is the tree with diameter 3 and central vertices of degrees a and b . Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise

Double-stars, two colors Definition The double-star S a , b is the tree with diameter 3 and central vertices of degrees a and b . Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof (lower bound). R ∆ ( S a , b ; 2 ) ≥ R ∆ ( K 1 , b ; 2 ) ; this suffices unless a = b and b is even.

Double-stars, two colors Definition The double-star S a , b is the tree with diameter 3 and central vertices of degrees a and b . Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof (lower bound). R ∆ ( S a , b ; 2 ) ≥ R ∆ ( K 1 , b ; 2 ) ; this suffices unless a = b and b is even. For that case, suppose H is ( 2 b − 2 ) -regular. Alternate red and blue along the edges of an Eulerian circuit. This avoids S b , b .

Double-stars, two colors Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof sketch (upper bound). Let H be triangle-free and ( 2 b − 1 ) -regular; we claim H → S b , b .

Double-stars, two colors Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof sketch (upper bound). Let H be triangle-free and ( 2 b − 1 ) -regular; we claim H → S b , b . Suppose some 2-edge-coloring of H avoids S b , b . Call a vertex majority-red if it lies on b red edges and majority-blue if it lies on b blue edges. WLOG, a plurality of vertices are majority-red.

Double-stars, two colors Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof sketch (upper bound). Let H be triangle-free and ( 2 b − 1 ) -regular; we claim H → S b , b . Suppose some 2-edge-coloring of H avoids S b , b . Call a vertex majority-red if it lies on b red edges and majority-blue if it lies on b blue edges. WLOG, a plurality of vertices are majority-red. No red edge joins majority-red vertices, so each red edge has a majority-blue endpoint. Each majority-red vertex lies on at least b red edges and each majority-blue vertex lies on at most b − 1, hence more majority-blue vertices than majority-red vertices.

Double-stars, two colors Theorem � 2 b − 2 , b even and a < b R ∆ ( S a , b ; 2 ) = . 2 b − 1 otherwise Proof sketch (upper bound). If a < b and b even, we can do better. Take C 5 , replace each vertex with an independent set of size b − 1, and replace each edge with K b − 1 , b − 1 . ◮ Each vertex is red, blue, or balanced. . . ◮ If we avoid S b − 1 , b , . ◮ no red edges joining red vertices . . . . ◮ no red edges joining red and . . balanced vertices ◮ no blue edges joining blue vertices ◮ no blue edges joining blue and ... . balanced vertices . .

Double-stars, many colors What about R ∆ ( S b , b ; s ) when s ≥ 3?

Double-stars, many colors What about R ∆ ( S b , b ; s ) when s ≥ 3? Theorem (Jiang) When T is a tree, we have R ∆ ( T ; s ) ≤ 2 s (∆( T ) − 1 ) . Thus R ∆ ( S b , b ; s ) ≤ 2 s ( b − 1 ) . Is this tight?

Double-stars, many colors What about R ∆ ( S b , b ; s ) when s ≥ 3? Theorem (Jiang) When T is a tree, we have R ∆ ( T ; s ) ≤ 2 s (∆( T ) − 1 ) . Thus R ∆ ( S b , b ; s ) ≤ 2 s ( b − 1 ) . Is this tight? Asymptotically yes, as s tends to infinity.

Double-stars, many colors Goal: extend upper bound argument for R ∆ ( S b , b ; 2 ) to R ∆ ( S b , b ; s ) for general s . Definition Given an s -edge-coloring of H : ◮ vertex v is major in a color if it lies on b edges of that color and minor otherwise; “Major in red” generalizes “majority-red” from the 2-color argument.

Double-stars, many colors Goal: extend upper bound argument for R ∆ ( S b , b ; 2 ) to R ∆ ( S b , b ; s ) for general s . Definition Given an s -edge-coloring of H : ◮ vertex v is major in a color if it lies on b edges of that color and minor otherwise; ◮ a minor edge is an edge whose color is minor at both endpoints. “Major in red” generalizes “majority-red” from the 2-color argument.

Double-stars, many colors Lemma Let H be a triangle-free graph. If an edge-coloring of H having r minor edges avoids S b , b , then � d ∗ ( v ) , | E ( H ) | + r = v where d ∗ ( v ) denotes the number of edges incident to and minor at v.

Double-stars, many colors Lemma Let H be a triangle-free graph. If an edge-coloring of H having r minor edges avoids S b , b , then � d ∗ ( v ) , | E ( H ) | + r = v where d ∗ ( v ) denotes the number of edges incident to and minor at v. Proof. No edge is major at both endpoints, since then it would be the central edge of a monochromatic S b , b . Count edges by the endpoint(s) at which they are minor; exactly r edges get counted twice.

Double-stars, many colors Lemma Let H be a triangle-free graph. If an edge-coloring of H having r minor edges avoids S b , b , then � d ∗ ( v ) , | E ( H ) | + r = v where d ∗ ( v ) denotes the number of edges incident to and minor at v. Proof. No edge is major at both endpoints, since then it would be the central edge of a monochromatic S b , b . Count edges by the endpoint(s) at which they are minor; exactly r edges get counted twice. Suppose we can find some H in which every edge-coloring avoiding S b , b has many minor edges. v d ∗ ( v ) can’t be too large, so we can bound | E ( H ) | , and hence ∆( H ) . � If ∆( H ) just barely exceeds this bound, then we’ve forced S b , b .

Double-stars, many colors Ramanujan graphs have the needed properties. Definition A Ramanujan graph is a d -regular graph with smallest eigenvalue at least √ − 2 d − 1.

Double-stars, many colors Ramanujan graphs have the needed properties. Definition A Ramanujan graph is a d -regular graph with smallest eigenvalue at least √ − 2 d − 1. Let H be a Ramanujan graph. No matter how we color E ( H ) , many edges will join vertices having the same major colors. These edges must be minor.