ATLANTA LECTURE SERIES IN COMBINATORICS AND GRAPH THEORY XIV

Transcription

1 ATLANTA LECTURE SERIES IN COMBINATORICS AND GRAPH THEORY XIV February 14-15, 2015 Georgia State University Department of Mathematics and Statistics

2 Featured Speaker Jozsef Balogh Department of Mathematical Sciences, University of Illinois One-hour Speakers Boris Bukh, Department of Mathematical Sciences, Carnegie Mellon University Shuhong Gao, Department of Mathematical Sciences, Clemson University Hao Huang, Institute for Mathematics and its Applications, University of Minnesota Chris Rodger, Department of Mathematics and Statistics, Auburn University Half-hour Speakers Carl Georg Heise, Institute of Mathematics, Hamburg University of Technology Bernard Lidický, Department of Mathematics, Iowa State University Colton Magnant, Department of Mathematical Sciences, Georgia Southern University Erik Westlund, Department of Mathematics, Kennesaw State University Shoichi Tsuchiya, Department of Mathematics, Keio University 1

3 Schedule Feb. 14 Speakers Topics 12:30-12:55pm Colton Magnant Forbidden properly colored subgraphs implying large highly connected monochromatic subgraphs 1:00-1: 50pm Jozsef Balogh On the applications of counting independent sets in hypergraphs I 2:00-2:50pm Chris Rodger Amalgamations of Graphs and Hypergraphs 2:50-3:20pm Break 3:20-4:10pm Hao Huang Biclique decomposition of random graphs 4:15-4:40pm Carl Georg Heise Coloring hypergraphs that are embeddable into R d 4:45-5:10pm Shoichi Tsuchiya On forbidden pairs for the existence of a s- panning Halin subgraph Feb. 15 Speakers Topics 8:30-8:55am Bernard Lidický Upper bounds on small Ramsey numbers 9:00-9:50am Jozsef Balogh On the applications of counting independent sets in hypergraphs II 9:55-10:45am Boris Bukh Algebraic constructions of Turán graphs 10:50-11:40pm Shuhong Gao Expander graphs and good linear codes 11:45-12:10pm Erik Westlund Precoloring Extensions using Hall s Condition 2

4 Forbidden properly colored subgraphs implying large highly connected monochromatic subgraphs Colton Magnant Highly connected subgraphs are extremely important to network applications of graph theory. When several colors are available for the edges, it is often difficult to force a monochromatic highly connected subgraph that is spanning. We discuss some recent results implying that an edge-coloring of a complete graph has an almost spanning monochromatic k-connected subgraph (ASMSk). The forbidden rainbow subgraphs that imply an ASMSk have recently been classified so we turn our attention to properly colored subgraphs. As should be expected, the set of proper subgraphs is much larger than the set of rainbow subgraphs. We will list some positive and negative results and outline a proof that the 6-cycle is among those forbidden proper subgraphs that yields an ASMSk. We wonder if every cycle with length a multiple of 3 is also in the set. 3

5 On the applications of counting independent sets in hypergraphs I-II Jozsef Balogh Recently, Balogh-Morris-Samotij and Saxton-Thomason developed a method of counting independent sets in hypergraphs. During the first talk, I will survey the field, and explain several applications. In the second talk I will show proof of several of these applications, which include counting maximal triangle-free graphs, maximal sum-free sets, intersecting families of permutations, intersecting families set-systems. These results are partly joint with Das, Delcourt, Liu, Morris, Mycroft, Petrickova, Samotij, Sharifzadeh and Treglown. 4

6 Amalgamations of Graphs and Hypergraphs Chris Rodger In this talk, recent uses of amalgamations of graphs will be described, starting with their applications to graphs, then seeing more recent use of the approach for hypergraphs. The results obtained provide both factorizations of (hyper)graphs and embeddings of edge-colored hyper(graphs). 5

7 Biclique decomposition of random graphs Hao Huang The biclique partition number bp(g) is the minimum number of complete bipartite graphs needed to partition the edges of a graph G. It is not hard to see that bp(g) n α(g), where α(g) is the independence number. Erdős conjectured that for the random graph G = G(n, 0.5), bp(g) = n α(g) with high probability. In this talk I will discuss some recent progress and and remaining challenges in this area, and show that actually there exists an absolute constant c > 0 such that for G = G(n, 0.5), bp(g) n (1 + c)α(g) with high probability. Joint work with Noga Alon and Tom Bohman. 6

8 Coloring hypergraphs that are embeddable into R d Carl Georg Heise Let H(d, k) be the set of all k-uniform hypergraphs that can be (linearly or piecewise linearly) embedded into R d. I will investigate lower and upper bounds on the maximum chromatic number of hypergraphs in H(d, k) depending on the parameters k and d and the number of vertices n. In particular, for d > 2, there is a construction for hypergraphs in H(2d 3, d) on n vertices whose weak chromatic number is Ω(log n/ log log n) and thus (choosing d = 3) a general 3D-version of the Four Color Theorem cannot hold. Also, the chromatic number for n-vertex hypergraphs in H(d, d) is bounded by O(n ( (d 2)/(d 1))) for d > 2. This is joint work with Oleg Pikhurko (University of Warwick), Konstantinos Panagiotou (Ludwig-Maximilians-Universität München) and Anusch Taraz (Hamburg University of Technology). 7

9 On forbidden pairs for the existence of a spanning Halin subgraph Shoichi Tsuchiya A Halin graph, defined by Halin [6], is a plane graph H = T C such that T is a spanning tree of H with no vertices of degree 2 where T 4 and C is a cycle whose vertex set is the set of leaves of T. In his work, as an example of a class of edge-minimal 3-connected plane graphs, Halin constructed this family of plane graphs. Although it was proved that Halin graphs satisfy many interesting properties, we know little about answers for the spanning Halin subgraph problem (i.e., which graph has a Halin graph as a spanning subgraph). In order to obtain some positive answers for the problem, we focus on forbidden subgraphs. For a set H of connected graphs, a graph G is said to be H-free if G contains no member of H as an induced subgraph. We also say that the members of H are forbidden subgraphs. When H = 2, H is called a forbidden pair. In graph theory, there are a lot of theorems on forbidden pairs that force some properties. For example, forbidden pairs for the existence of a Hamilton cycle in 2-connected graphs were studied in [2, 3, 5], and a characterization of such pairs was accomplished by Bedrosian [1]. Later, Faudree and Gould [4] extended the result of Bedrossian by regarding finite number of 2-connected {H 1, H 2 }-free non-hamiltonian graphs as exceptions. In this talk, we introduce recent results on forbidden pairs for the existence of a spanning Halin subgraph in 3-connected graphs. References [1] P. Bedrossian, Forbidden subgraph and minimum degree conditions for Hamiltonicity, Ph.D. Thesis, Memphis State University, [2] H.J. Broersma and H.J. Veldman, Restrictions on induced subgraphs ensuring Hamiltonicity or pancyclicity of K 1,3 -free graphs, in: R. Bodendiek, ed., Contemporary Methods in Graph Theory (Mannheim, 1990) [3] D. Duffus, R.J. Gould and M.S. Jacobson, Forbidden subgraphs and the hamiltonian theme, The theory and applications of graphs, , Wiley, New York,

10 [4] R.J. Faudree and R.J. Gould, Characterizing forbidden pairs for Hamiltonian properties, Discrete Math. 173 (1997) [5] R.J. Gould and M.S. Jacobson, Forbidden subgraphs and Hamiltonian properties of graphs, Discrete Math. 42 (1982), [6] R. Halin, Studies on minimally n-connected graphs, Combinatorial mathematics and its applications, edited by D. J. A. Welsh (Academic Press, New York, 1971),

11 Upper bounds on small Ramsey numbers Bernard Lidický We show how to compute upper bounds on small Ramsey numbers using flag algebras and we obtain improvements for several cases. Although flag algebras were developed for investigating large graphs, it is possible to use them for small graphs by considering blowups. The nice feature of the flag algebra approach is that it allows to attack upper bounds for different Ramsey numbers with only minor modifications. This is joint work with Florian Pfender. 10

12 Algebraic constructions of Turán graphs Boris Bukh Pick a bipartite graph H. What is the graph with the largest number of edges and no copy of H? Evidence hints at an algebraic answer to this question. In this talk, I will present the available evidence, and explain the current obstacles. 11

13 Expander graphs and good linear codes Shuhong Gao Expander graphs are highly connected sparse finite graphs. They play an important role in several areas of mathematics, including number theory (e.g. sieves for primes in group orbits), representation theory and geometric embeddings, as well as in computer sciences and digital signal processing, including communications network designs, pseudorandom number generators, compressive sensing, algorithm designs, among others. In this talk, we present a survey of expander graphs (including Ramanujan graphs), their connections to good linear codes and recent progress on fast decoding algorithms. 12

14 Precoloring Extensions using Hall s Condition Erik Westlund In 1990 Hilton and Johnson introduced Hall s condition, a generalization of Hall s Marriage Theorem applied to list assignments of the vertex set of a graph. A list assignment L to a graph G is called Hall if (G, L) satisfy Hall s condition, which is necessary (but not always sufficient) for G to admit a proper L-coloring. A graph G is Hall m-completable if every proper m-precoloring of G, whose corresponding list assignment is Hall, can be extended to a proper m-coloring of G. In 2011, Bobga et al. asked if a graph G is always Hall (G)- completable, thereby posing a possible precoloring extension version of Brooks theorem. We give a straightforward and short proof that answers this question in the affirmative as well as discuss some related results and open questions. Additionally, for several graph families, we will examine the full spectrum of values for which a graph is Hall m-completable. This yields some surprising results, such as graphs which are both Hall m-completable and Hall (m + k)-completable, for infinitely many k > 1, but are not Hall (m + 1)-completable. This is joint work with Sarah Holliday and Jennifer Vandenbussche. 13