Posts Tagged ‘TommyJensen’

Special session on graph theory, 2011 Spring Meeting of the Korean Mathematical Society

Saturday, March 26th, 2011
Special Session on Graph Theory – 2011 spring Meeting of the Korean Mathematical Society
April 30, 2011, 9:00-11:40
Asan Science Building (아산이학관), Korea University (고려대), Seoul

Preregistration deadline: April 11

  • 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
Sang-il Oum (엄상일)
Department of Mathematical Sciences, KAIST
We prove that every infinite sequence of skew-symmetric or symmetric matrices M1, M2, … over a fixed finite field must have a pair Mi, Mj (i<j) such that that Mi is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in Mj, 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
Sejeong Bang (방세정)
Department of Mathematics, Yeungnam University
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(a1+1)/3}<k<4a1+10−6c2. 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(a1+1)/3}<k<4a1+10−6c2 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
Mark H. Siggers
Department of Mathematics, Kyungpook National University
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
Tommy R. Jensen
Department of Mathematics, Kyungpook National University
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
Jack Koolen
Department of Mathematics, POSTECH
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.

Tommy R. Jensen, The 3-Color Problem

Friday, January 15th, 2010
The 3-Color Problem
Tommy R. Jensen
Department of Mathematics, Kyungpook National University, Daegu, Korea
2010/02/19 Friday 4PM-5PM

The fundamental sufficient condition for the existence of a proper 3-coloring of the vertices of a planar graph G was proved by Grötzsch more than 50 years ago, and it requires that G has no triangles (cycles of length 3). This talk discusses conjectures for other possible sufficient conditions, some of which have stubbornly resisted proofs for decades, and also various recent partial results. A conjecture in a different direction deals with a stronger 3-colorability property, which for a planar graph turns out to be equivalent to triangle-freeness, but here it is unknown whether the assumption of planarity may be weakened.

Tommy R. Jensen, The Cycle Double Cover Problem for graphs

Wednesday, April 8th, 2009
The Cycle Double Cover Problem for graphs
Tommy R. Jensen
Department of Mathematics, Kyungpook National University, Daegu, Korea
2009/04/24 Friday 4PM-5PM

The Cycle Double Cover Problem in Graph Theory suggests that all 2-connected graphs share a certain property with 2-connected planar maps. Such a map clearly contains a collection of cycles, indeed the boundary cycles of its faces, such that each edge belongs to exactly
two of them. The generalization of this property to nonplanar graphs remains one of the central open problems in Graph Theory.

We investigate this problem by generalizing a suitable variation of the statement of another almost obvious property of planar maps, namely the Jordan Curve Theorem. The generalization suggests a new conjecture which is much stronger than the Cycle Double Cover Conjecture. In fact it would imply a very strong form of the Cycle Double Cover Conjecture, suggesting that every cycle in a 2-connected graph appears in at least one cycle double cover of the graph.

We prove the stronger conjecture in a few important special cases.