April 30, 2011, 9:00-11:40

Preregistration deadline: April 11

Timetable

- 9:00-9:30 Sang-il Oum (엄상일), KAIST : Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices
- 9:30-10:00 Sejeong Bang (방세정), Yeungnam University : Geometric distance-regular graphs with smallest eigenvalue -3
- 10:00-10:10 Break
- 10:10-11:40 Mark H. Siggers, Kyungpook National University : The H-colouring Dichotomy through a projective property
- 10:10-10:40 Tommy R. Jensen, Kyungpook National University : On second Hamilton circuits in cubic graphs
- 11:10-11:40 Jack Koolen, POSTECH : Recent progress of distance-regular graphs

Organized by Seog-Jin Kim (Konkuk University) and Sang-il Oum (KAIST).

At 14:00-14:40, there will be an invited talk by Xuding Zhu, *Thue choice number of graphs*.

Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_{1}, M_{2}, … over a fixed finite field must have a pair M_{i}, M_{j} (i<j) such that that M_{i} is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_{j}, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour’s theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle’s theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivot-minors.

Geometric distance-regular graphs with smallest eigenvalue −3

A non-complete distance-regular graph Γ is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique clique in C. In this talk, we determine the non-complete distance-regular graphs satisfying max{3,8(a_{1}+1)/3}<k<4a_{1}+10−6c_{2}. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying max{3,8(a_{1}+1)/3}<k<4a_{1}+10−6c_{2} is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, 7 feasible intersection arrays are ruled out.

The H-colouring Dichotomy through a projective property

The H-colouring Dichotomy of Hell and Nesetril, proved in 1990, is one of the most quoted results in the field of Graph Homomorphisms. It says that H-coloring, the problem of deciding if a given graph G admits an homomorphism to the fixed graph H, is NP-complete if H contains an odd cycle, and otherwise polynomial time solvable.

In this talk we present a short new proof of this result, recently published, using a new projective property defined for homomorphisms of powers of a graph G onto a graph H.

On second Hamilton circuits in cubic graphs

A classical theorem of Cedric Smith guarantees the existence of a second Hamilton circuit other than a given one in any hamiltonian cubic graph. It is an open problem in complexity theory whether the corresponding search problem is polynomially solvable. We observe that a search algorithm, implicit in Bill Tutte’s nonconstructive proof of Smith’s theorem, has exponential running time. We also mention two possible candidates for search algorithms with polynomial complexity.

Recent progress of distance-regular graphs

I will talk about recent progress of distance-regular graphs.

(Invited lecture at 2PM)

Thue choice number of graphs

Xuding Zhu
Institute of Mathematics, Zhejiang Normal University, Jinhua, China

A sequence of even length is a repetition if the first half is identical to the second half. A sequence is said to contain a repetition if it has a subsequence which is a repetition. A classical result of Thue says that there is an infinite sequence on 3 symbols which contains no repetition. This result motivated many deep research and challenging problems. One graph concept related to this result is Thue-colouring. A Thue-colouring of a graph G is a mapping which assigns to each vertex of G a colour (a symbol) in such a way that the colour sequence of any path of G contains no repetition. The Thue-chromatic number of a graph is the minimum number of colours needed in a Thue-colouring of G. Thue’s result is equivalent to say that the infinite path has Thue-chromatic number 3. It is also known that the Thue-chromatic number of any tree is at most 4.

Thue-choice number of a graph G is the list version of its Thue-chromatic number, which is the minimum integer k such that if each vertex of G is given k-permissible colours, then there is a Thue-colouring of G using a permissible colour for each vertex. This talk will survey some research related to Thue Theorem and will show that Thue-choice number of paths is at most 4 and Thue choice number of trees are unbounded.