T´om la.i, o’ chu.o.ng n`. ay ch´ung ta d¯˜a chı’ ra d¯u.o.. c d¯ieˆ`u kie.ˆn d¯u’ cho su.. toˆ`n ta.i chu tr`ınh Hamilton trong mo.ˆt soˆ´ d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4. C´ac d¯i.nh l´y 3.6, 3.7, 3.8 v`a 3.9 l`a nh˜u.ng keˆ´t qua’ nghieˆn c´u.u treˆn c´ac
d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4 lieˆn thoˆng c´o bieˆ’u tu.o..ng th´u. nhaˆ´t kh´ac roˆ˜ng. D- i.nh l´y 3.10 d¯˜a x´et d¯eˆ´n c´ac d¯oˆ` thi. n`ay khi ch´ung c´o bieˆ’u tu.o..ng th´u. nhaˆ´t b˘a`ng roˆ˜ng. O’ d¯aˆy, ch´ung ta d¯˜a su.’ du.ng t´o.i mo.ˆt v`ai keˆ´t qua’. tru.´o.c d¯aˆy veˆ` d¯oˆ` thi. Cayley, d¯oˆ` thi. thu.o.ng, d¯oˆ` thi. Petersen toˆ’ng qu´at
GP(n, k) v`a c´ac k˜y thua.ˆt cu’a toˆ’ ho. p d. ¯eˆ’ xaˆy du.. ng tru. c tie. ˆ´p chu tr`ınh Hamilton trong d¯oˆ` thi..
Tieˆ´p tu.c mo’ ro.ˆng nghieˆn c´u. .u theo hu.´o.ng n`ay, hy vo.ng r˘a`ng trong th`o.i gian t´o.i, ch´ung ta s˜e c´o theˆm nh˜u.ng keˆ´t qua’ saˆu s˘a´c ho.n treˆn l´o.p d¯oˆ` thi. d¯ang x´et.
Taˆ´t ca’ c´ac keˆ´t qua’ d¯u.o..c tr`ınh b`ay trong lua.ˆn ´an d¯eˆ`u xoay quanh l´o.p d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4, d¯oˆ´i tu.o..ng ch´ınh trong lua.ˆn ´an. V´o.i hai mu.c d¯´ıch d¯˘a.t ra l`a x´et t´ınh lieˆn thoˆng v`a su. to. ˆ`n ta.i chu tr`ınh Hamilton cu’a c´ac d¯oˆ` thi. n`ay, lua.ˆn ´an d¯˜a d¯a.t d¯u.o..c c´ac keˆ´t qua’ sau d¯aˆy:
1. Xaˆy du.. ng k˜y thua.ˆt toˆ’ng qu´at d¯eˆ’ x´ac d¯i.nh d¯ieˆ`u kie.ˆn lieˆn thoˆng cho c´ac d¯oˆ` thi. meta luaˆn ho`an n´oi chung. K˜y thua.ˆt n`ay d¯u.o..c theˆ’ hie.ˆn trong c´ac me.ˆnh d¯eˆ` 2.1, 2.2 v`a 2.3 c`ung v´o.i vie.ˆc ´ap du.ng Me.ˆnh d¯eˆ` 1.1.
2. Su.’ du.ng k˜y thua.ˆt n´oi treˆn, lua.ˆn ´an d¯˜a chı’ ra d¯u.o..c d¯ieˆ`u kie.ˆn caˆ`n v`a d¯u’ cho t´ınh lieˆn thoˆng cu’a d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4. D- i.nh l´y 2.5 l`a c´ac d¯ieˆ`u kie.ˆn d`anh cho c´ac d¯oˆ` thi. c´o bieˆ’u tu.o..ng th´u. nhaˆ´t kh´ac roˆ˜ng. Khi c´ac d¯oˆ` thi. c´o bieˆ’u tu.o..ng th´u. nhaˆ´t b˘a`ng roˆ˜ng, ch´ung ta c´o D-i.nh l´y 2.11. Ngo`ai ra, mo.ˆt thu’ tu.c kieˆ’m tra t´ınh lieˆn thoˆng cu’a ch´ung du. a v`. ao hai d¯i.nh l´y treˆn c˜ung d¯u.o..c d¯eˆ` xuaˆ´t.
3. Veˆ` vaˆ´n d¯eˆ` Hamilton, lua.ˆn ´an d¯˜a ch´u.ng minh d¯u.o..c c´ac d¯i.nh l´y 3.6, 3.7, 3.8 v`a 3.9 veˆ` d¯ieˆ`u kie.ˆn d¯u’ cho su. to. ˆ`n ta.i chu tr`ınh Hamilton trong c´ac d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4 c´o bieˆ’u tu.o..ng th´u. nhaˆ´t kh´ac roˆ˜ng. Khi bieˆ’u tu.o..ng th´u. nhaˆ´t cu’a ch´ung b˘a`ng roˆ˜ng, D- i.nh l´y 3.10 d¯˜a chı’ ra d¯u.o.. c mo.ˆt v`ai d¯ieˆ`u kie.ˆn d¯eˆ’ d¯oˆ` thi. M C(2, n, α, S0, S1) v´o.i |S1| = 4 c´o chu tr`ınh Hamilton.
Tieˆ´p tu.c nghieˆn c´u.u veˆ` chu tr`ınh Hamilton trong d¯oˆ` thi. meta luaˆn ho`an ba.ˆc 4, ch´ung toˆi d¯ang xem x´et t´o.i c´ac d¯oˆ` thi. c´o m chia heˆ´t cho 4 v`a d¯˜a thu d¯u.o.. c nh˜u.ng keˆ´t qua’ ban d¯aˆ`u. Hy vo.ng r˘a`ng trong th`o.i gian t´o.i, ch´ung toˆi s˜e gia’i quyeˆ´t d¯u.o.. c tro.n ve.n tru.`o.ng ho..p n`ay.
C´ac coˆng tr`ınh d¯˜a coˆng boˆ´ c´o lieˆn quan d¯eˆ´n lua.ˆn ´an:
1. Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”, In: Proceed- ings of The International Conference “Mathematical Foundation of Informatics” (October 25 – 28, 1999, Hanoi, Vietnam), World Sci- entific, Singapore (nha.ˆn d¯˘ang).
2. Ngo Dac Tan and Tran Minh Tuoc, “On Hamilton cycles in con- nected tetravalent metacirculant graphs with non-empty first sym- bol”, Acta Mathematica Vietnamica 28 (2003), 267 - 278.
3. Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetrava- lent metacirculant graphs with the empty first symbol”, Preprint 2002/33, Institute of Mathematics (2002) (gu.’ i d¯˘ang).
C´ac t´om t˘a´t b´ao c´ac ta.i c´ac ho.ˆi nghi.:
1. Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”,Abstract of The International Conference “Mathematical Foundation of Informat- ics”, October 25 – 28, 1999, Hanoi, Vietnam.
2. Ngo Dac Tan and Tran Minh Tuoc, “On Hamilton cycles in con- nected tetravalent metacirculant graphs with non-empty first sym- bol”, Abstract of The International Conference “Combinatorics and Applications”, December 03 – 05, 2001, Hanoi, Vietnam.
3. Ngoˆ D- ˘a´c Taˆn v`a Traˆ`n Minh Tu.´o.c, “Connectedness of tetravalent metacirculant graphs with the empty first symbol”, T´om t˘a´t c´ac b´ao c´ao Ho.ˆi nghi. To´an ho. c To`an quoˆ´c laˆ`n th´u. 6, 07 – 10/09/2002,
Hueˆ´, Vie.ˆt Nam.
[1] Alspach B. (1983), “The classification of hamiltonian generalized Petersen graphs”, J. Combin. Theory Ser. B 34, 293 - 312.
[2] Alspach B. (1989), “Lifting Hamilton cycles of quotient graphs”,
Discrete Mathematics 78, 25 - 36.
[3] Alspach B. (1989), “Hamiton cycles in metacirculant graphs with prime cardinal blocks”, Annals of Discrete Mathematics Vol. 41, 25 - 36.
[4] Alspach B., Durnberger E. and Parsons T.D. (1985), “Hamiton cy- cles in metacirculant graphs with prime cardinality blocks”, Annals of Discrete Mathematics 27, 27 - 34.
[5] Alspach B. and Parsons T.D. (1982), “A construction for vertex- transitive graphs”, Canad. J. Math. 34, 307 - 318.
[6] Alspach B. and Parsons T.D. (1982), “On Hamilton cycles in metacirculant graphs”, Ann. of Discrete Math. 15, 1 - 7.
[7] Alspach B. and Sutcliffe R.J. (1979), “Vertex-transitive graphs of order 2p”, in: Ann. New York Acad. Sci., New York Acad. Sci., 18 - 27.
[8] Alspach B. and Zhang C.Q. (1989), “Hamilton cycles in cubic Cay- ley graphs on dihedral groups”, Ars Combin. 28, 101 - 108.
[9] Behzad M. and Chartrand G. (1971), Introduction to the theory of graphs, Allyn and Bacon, Boston.
[10] Bermond J.C. (1978), “Hamitolnian graphs”, Selected topics in Graphs Theory, Academic Press, London.
[11] Biggs N. (1973), “Three remarkable graphs”, Canadian Journal of Mathematics 25, 397 – 411.
[12] Bollob´as B. (1979),Graph Theory, an introductory course, Springer (adsbygoogle = window.adsbygoogle || []).push({});
- Verlag, New York.
[13] Diestel R. (2000), Graph Theory, Springer, Berlin.
[14] Dirac G.A. (1952), “Some theorem on Abstract graphs”,Proc. Lon- don Math. Soc. 2, 69 - 81.
[15] Dobson E. (1998), “Isomorphism problem for metacirculant graphs of order a product of two primes”, Canad. J. Math.50, 1176 - 1188. [16] Durnberger E. (1983), “Connected Cayley graphs of semi-direct product of cyclic groups of prime order by abelian groups are Hamil- tonian”, Discrete Mathematics 46, 55 - 68.
[17] Godsil C. and Royle G.F. (2001), Algebraic graph theory, Springer
- Verlag, New York.
[18] Gould R.J. (1991), “Updating the hamiltonian problem, a survey”,
J. Graph Theory 15, 121-157.
[19] Gross J., Yellen J. (1999), Graph Theory and its applications, CRC
Press, Boca Raton - London - New York - Washington.
[20] Lipman M.J. (1985), “Hamilton cycles and paths in vertex- transitive graphs with abelian and nilpotent groups”, Discrete Mathematics 54, 15 - 21.
[21] Lov´asz L. (1970),Combinatorial Structures and Their Applications,
Gordon and Breach, London Problem II.
[22] Maruˇsiˇc D. (1983), “Hamilton circuits in Cayley graphs”, Discrete Mathematics 46, 49 - 54.
[23] Maruˇsiˇc D. and Scapellato R. (1994), “Classifying vertex-transitive graphs whose order is a product of two primes”, Combinatorica 14, 184 - 201.
[24] Ore O. (1960), “A note on Hamiltonian circuits”, Am. Math. Month. 67, 55.
[25] Ngo Dac Tan (1990), “On cubic metacirculant graphs”,Acta Math- ematica Vietnamica Vol. 15, No. 2, 57 - 71.
[26] Ngo Dac Tan (1992), “Hamilton cycles in cubic (4, n)-metacirculant graphs”, Acta Math. Vietnamica Vol. 17, No. 2, 83 - 93.
[27] Ngo Dac Tan (1993), “Connectedness of cubic metacirculant graphs”, Acta Math. Vietnamica Vol. 18, No.1, 3 - 17.
[28] Ngo Dac Tan (1993), “On Hamilton cycles in cubic (m, n)- metacirculant graphs”, Australasian Journal of Combinatorics 8, 211 - 232.
[29] Ngo Dac Tan (1994), “Hamilton cycles in cubic (m, n)- metacirculant graphs with m divisible by 4”, Graphs and Combin.
10, 67 - 73.
[30] Ngo Dac Tan (1995), “Hamilton cycles in some vertex-transitive graphs”, SEA Bull. Math. Vol. 19, No. 1, 61 - 67.
[31] Ngo Dac Tan (1996), “On Hamilton cycles in cubic (m, n)- metacirculant graphs, II”, Australasian Journal of Combinatorics
14, 235 - 257.
[32] Ngo Dac Tan (1996), “Non-Cayley tetravalent metacirculant graphs and their hamiltonicity”, Journal of Graph Theory 23, 273 - 287. [33] Ngo Dac Tan (1996), “On the isomorphism problem for a family of
[34] Ngo Dac Tan (1996), “Cubic (m, n)-metacirculant graphs which are not Cayley graphs”, Discrete Mathematics 154, 237 - 244.
[35] Ngo Dac Tan (1997), “Sufficient conditions for the existence of a Hamilton cycles in cubic (6, n)-metacirculant graphs”, Vietnam Journal of Mathematics 25:1, 41 - 52.
[36] Ngo Dac Tan (1998), “Sufficient conditions for the existence of a Hamilton cycles in cubic (6, n)-metacirculant graphs, II”, Vietnam Journal of Mathematics 26:3, 41 - 52.
[37] Ngo Dac Tan (2002), “On Non-Cayley tetravalent metacirculant graphs”, Graphs and Combin. 18, 795 - 802. (adsbygoogle = window.adsbygoogle || []).push({});
[38] Ngo Dac Tan (2003), “The automorphism groups of certain tetrava- lent metacirculant graphs”, Ars Combinatoria 66, 205 - 232.
[39] Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”, In: Proceed- ings of The International Conference “Mathematical Foundation of Informatics” (October 25 – 28, 1999, Hanoi, Vietnam), World Sci- entific, Singapore. (Nha.ˆn d¯˘ang)
[40] Ngo Dac Tan and Tran Minh Tuoc (2003), “On Hamilton cycles in connected tetravalent metacirculant graphs with non-empty first symbol”, Acta Mathematica Vietnamica 28, 267 - 278.
[41] Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with empty first symbol”, Preprint 2002/33,
Institute of Mathematics. (Gu.’ i d¯˘ang)
[42] Wielandt H. (1964), Finite permutation groups, Academic Press,