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.
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