The 57th KPPY Combinatorics Seminar Organized by S.Bang, M.Hirasaka, T.Jensen, J.Koolen and M.Siggers Jun 08 2013 Natural Sciences Building, Room 313 Department of Mathematics, Kyungpook National University
Program 11:00 - 11:50 Naoki Matsumoto Yokohama National University The size of edge-critical uniquely colorable planar graphs 1:30 -2:20 Jongyook Park University of Science and Technology of China On symmetric association schemes with mi = 3 2:30 -3:20 Sang June Lee KAIST On a Cameron–Erd˝os problem of Sidon sets. 3:40 - 4:30 Phan Thanh Toan POSTECH Improved Semidefinite Programming Bound on Sizes of Codes 4:40 - 5:30 Andreas Holmsen KAIST An oriented matroid version of the colorful Caratheodory theorem 6:00–8:00 Banquet
Abstracts Naoki Matsumoto The size of edge-critical uniquely colorable planar graphs A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G − e is not a uniquely k-colorable graph for any edge e ∈ E(G). In this paper, we prove that if G is an edge-critical uniquely . On the other hand, 3-colorable planar graph, then |E(G)| ≤ 38 |V (G)| − 17 3 there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with n vertices and 94 n − 6 edges. Our result gives a first non-trivial upper bound for |E(G)|.
Jongyook Park On symmetric association schemes with mi = 3 In 2006, Bannai and Bannai classified primitive symmetric association schemes with m1 = 3. In their paper, the hardest case was k1 = 3. Even though, (primitive) symmetric association schemes with k1 = 3 were classified by Yamazaki in 1998, they avoided the use of the difficult and deep result of Yamazaki. In this talk, we generalize the result of Bannai and Bannai. This is joint work with M. Camara, E. R. van Dam and J. H. Koolen.
Sang June Lee On a Cameron–Erd˝os problem of Sidon sets. A set A of positive integers is called a Sidon set if all the sums a1 + a2 , with a1 ≤ a2 and a1 , a2 ∈ A, are distinct. In this talk we consider Cameron– Erd˝os problem which was suggested in 1990. The problem is to estimate the number of Sidon sets contained in [n] := {1, 2, . . . , n}. Results of Chowla, Erd˝os, Singer, and √ Tur´an from the 1940s imply that the maximum size of Sidon sets in [n] is n(1 + o(1)). From this result, one can trivially√obtain that the number of Sidon sets contained in [n] is between √ (1+o(1)) n c n log n and 2 for some absolute constant c. However, these bounds 2 have not been notably improved for about 20 years. √ We obtain a new upper bound 2c n which is sharp up to a constant√ factor in the exponent when compared to the previous lower bound 2(1+o(1)) n . For
the proof, we define a graph from our setting such that, roughly speaking, a Sidon set in [n] corresponds to an independent set of the graph. In addition, the graph satisfies some dense condition. We show that in a graph satisfying the dense condition, the number of independent sets of given size t is much smaller than the trivial bound nt . By applying it repeatedly, we have the new upper bound on the number of Sidon sets contained in [n]. This is joint work with Kohayakawa, R¨odl, and Samotij.
Phan Thanh Toan Improved Semidefinite Programming Bound on Sizes of Codes Let A(n, d) (respectively A(n, d, w)) be the maximum possible number of codewords in a binary code (respectively binary constant-weight w code) of length n and minimum Hamming distance at least d. By adding new linear constraints to Schrijver’s semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on A(n, d), namely A(18, 8) ≤ 71 and A(19, 8) ≤ 131. Twenty three new upper bounds on A(n, d, w) for n ≤ 28 are also obtained by a similar way. This is a joint work with Prof. Hyun Kwang Kim.
Andreas Holmsen An oriented matroid version of the colorful Caratheodory theorem We give the following extension of Barany’s colorful Caratheodory theorem: Let M be an oriented matroid and N a matroid with rank function r, both defined on the same ground set V and satisfying rank(M) < rank(N). If every subset A of V with r(V − A) < rank(M) contains a positive circuit of M , then some independent set of N contains a positive circuit of M .
