论文信息 - On minimum vertex covers of generalized Petersen graphs

On minimum vertex covers of generalized Petersen graphs

For natural numbers n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and subscripts are reduced modulo n. Here first, we characterize minimum vertex covers in generalized Petersen graphs. Second, we present a lower bound and some upper bounds for β(P (n, k)), the size of minimum vertex cover of P (n, k). Third, in some cases, we determine the exact values of β(P (n, k)). Our conjecture is that β(P (n, k)) ≤ n+ ⌈ 5 ⌉, for all n and k. 254 B. BEHSAZ, P. HATAMI AND E. S. MAHMOODIAN

[1] Ebadollah S. Mahmoodian,et al. Vertex domination of generalized Petersen graphs , 2009, Discret. Math..

[2] Pooya Hatami,et al. On minimum vertex cover of generalized Petersen graphs , 2010, ArXiv.

[3] R. Bruce Richter,et al. On the crossing numbers of certain generalized Petersen graphs , 1992, Discret. Math..

[4] Allen J. Schwenk. Enumeration of Hamiltonian cycles in certain generalized Petersen graphs , 1989, J. Comb. Theory, Ser. B.

[5] Kathryn Fraughnaugh,et al. Introduction to graph theory , 1973, Mathematical Gazette.

[6] G. Prins,et al. Every generalized Petersen graph has a Tait coloring. , 1972 .

[7] M. Watkins,et al. A theorem on tait colorings with an application to the generalized Petersen graphs , 1969 .

[8] Gelasio Salazar. On the crossing numbers of loop networks and generalized Petersen graphs , 2005, Discret. Math..

[9] Brian Alspach. The classification of hamiltonian generalized Petersen graphs , 1983, J. Comb. Theory, Ser. B.

[10] Kozo Bannai. Hamiltonian cycles in generalized petersen graphs , 1978, J. Comb. Theory, Ser. B.

[11] Clifford Stein,et al. Introduction to Algorithms, 2nd edition. , 2001 .