Hamiltonian Properties in
Cartesian Product
Mei Lu
Department of Mathematical Sciences,
Tsinghua University, Beijing
A joint work with H.J. Lai and H. Li
Terminology and notation
Let G=(V, E) be a simple graph.
dG(v): the degree of v in G.
Δ(G): the maximum degree of G
Pm: a path with m vertices (m≥2)
Cn: a cycle with n vertices (n≥3)
K1,j-1:a star with j vertices (j≥1)
Cartesian product of G1 and G2
Problem
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
Cn×Cm of two directed cycles to be Hamiltonian.
Namely, Cn×Cm is Hamiltonian if and only if there
exist positive integers d1 and d2 such that
gcd(d1, n) = 1 = gcd(d2,m) and d = d1 +d2 where
d = gcd(m, n).
Some known results
V.V. Dimakopoulos, L. Palios, A.S. Poulakidas, On the
hamiltonicity of the cartesian product, Information
Processing Letters 96 (2005) 49–53.
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.
Lemma
Theorem 1
Sketch of the proof
Results (2)
Theorem 3
Results (3)
Theorem 4
Lemma
Results (4)
Theorem 5
Results (5)
Theorem 6
Thank You！