1. Hamiltonian Properties in Cartesian Product Mei Lu Department of Mathematical Sciences, Tsinghua University, Beijing A joint work with H.J. Lai and H. Li

2. Terminology and notation Let G=(V, E) be a simple graph. d G (v): the degree of v in G. Δ(G): the maximum degree of G P m : a path with m vertices (m≥2) C n : a cycle with n vertices (n≥3) K 1,j-1 :a star with j vertices (j≥1)

3. Cartesian product of G 1 and G 2

5. Problem

6. In 1978, W. T. Trotter, Jr. and P. Erdos [J. Graph Theory 2 (1978), no. 2, 137–142;] gave necessary and sufficient conditions for the Cartesian product C n ×C m of two directed cycles to be Hamiltonian. Namely, C n ×C m is Hamiltonian if and only if there exist positive integers d 1 and d 2 such that gcd(d 1, n) = 1 = gcd(d 2,m) and d = d 1 +d 2 where d = gcd(m, n).

7. Some known results

8. V.V. Dimakopoulos, L. Palios, A.S. Poulakidas, On the hamiltonicity of the cartesian product, Information Processing Letters 96 (2005) 49–53.

9. Results (1) Theorem 1 Corollary 2 V. Batagelj and T. Pisanski, Hamiltonian cycles in the Cartesian product of a tree and a cycle, Discrete Math. 38 (1982), no. 2-3, 311– 312.

10. Lemma

11. Theorem 1 Sketch of the proof

13. Results (2) Theorem 3

14. Results (3) Theorem 4

15. Lemma

16. Results (4) Theorem 5

18. Results (5) Theorem 6

19. Thank You ！