A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Veldman, H.J.

English

University of Twente Research Information

Discrete Mathematics 44 (1983) 309-3i6 309 North-Holland Publishing Company E X I S T E N C E  O F  D . , , - , C Y C L E S  A N D  D x - P A T H S  H.J.  V E L D M A N  Department of A~plied Mathematics, Twente University of Technology, Enschede, the Nether- lands Received 4 March 1982 Revised 28 May 1982 A cycle C of a graph G is called a D A-cycle if every component of G - V(C) has order less than A. A Dx-path is defined analogously. In particular, a Dl-cycle is a hamiltonian cycle and a Dl-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dx-cycle or Dx-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertey degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way 1. Introduction W e  employ  the terminology of Bondy and Murty  [3] and consider  only simple graphs. In [2], Bondy s ta ted a sufficient condi t ion for a graph G to have a cycle C such that  G - V ( C )  contains no Kk. Fo r  k = 1, it coincides with O r e ' s  condit ion for the existence of a hamil tonian cycle. He re  we int roduce another  kind of general ized hamil tonian cycle. A cycle C of a graph G is a D~-cycle if all components  of G - V ( C )  have order  less than A. Al ternat ively ,  C is a Dx-cycle of G if and only if every connected  subgraph of o rder  A of G has at least one vertex with C in common.  Thus a Dx-cycle  domina tes  all connected  subgraphs of o rder  ~. Ana log-  ously,  a path  P of  G is a D~-path if every component  of G - V(P)  has order  less than A. Graphs  containing a Dx-cycle  (Dx-path)  will be  called Dx-cyclic (Dx- traceable). A Dl -cyc le  (Dl -pa th )  is the same as a hamil tonian cycle (hamiltonian path).  D2-cycles were s tudied in [6]. In subsequent  sections, existence theorems for Dx-cycles are proved.  In [6], most  of them were a l ready proved for A = 2. W~ will hencefor th  refrain from referr ing to these special cases, unless this is essential.  Parallel  results on Dx-paths  can be obta ined,  using the following obvious lemma.  L e m m a  1. A graph G is Dx-traceable i[ and only i[ G v K ~  is Dx-cyciic. The  theorems der ived  are general izat ions of known result.,; in hamil tonian graph theory.  A corresponding r emark  can be made about  the proof  techniques used. Some of the results in Section 3 are closely re la ted  to Bondy ' s  work [2]. 0012-365XI8310000-0000/$03.00 © 1983 Nor th -Hol l and  310 H,J, Veldmrn Exte~,,sions to subgraphs of order greater than 1 of concepts such ab adjacency of vertices, independence number and vertex degree arise in correspondence with the generalization of hamiltonim~ cycles to D,~-cycles, 2. A necessary condition in ~erms of cu~ sets +I'0 stt~,l't with, we generalize a necessary condition for the existence of ~.~. hamiltonia,~ ,.'vole, ~heorem A 13, Theorem 4,21. Ii" (~ ,~r~l,.Iv ( ;  i,~ l~,~M;i~mi,,,,, flyers, ./'or eeery ¢'~((; ,~"<" I,% I)~.',,~ote bv ~o~ ( ( ; l  the ntlnll'~cr of .--onlpotletlts ,,)f ( i  ~1 order i~t lei,tsl A, Th@v,t'etll ,,\ is then a special case (A'  I] oF Theorem I .  I f  ~ ,~r~l)le ( ;  i.~ !)~-cyclic, Ilwn, for ~'rerv ~l~meml)ly pn~),'r .~d~.wl S of ~.'((;L Th,; pr+.~ol, b¢in.e, an c~tsy extension of the proof of I6, Theorem ~I. is omil tcd, l,'or f~lltlt't~, refer~.,n~.'e we t|ellote bv ;)C,~ the ~.'hlss oP' graphs t~.~ salisfyil|g the nccc,,,s~ry condition of Theorem I, Thus ( i  is in ,/~.~ i1!', for some u o ~  ~ p l y  proper .,,tll'~s~,,t ~ o!' ~'((;).  ,.,,~((i ,~'I~-I,~I , 3. Sufficient conditions involving subllraph dellrees We now turn our attentinn h~ sutlicicnt conditions for rite cxislcnce of l)~-¢yclcx, (Inc ~l' the earliest res:dts ill hanliltoni;m graph theory to be gcncralitcd hcl"c is due !o l)ira¢. ' rh~wem B j3, Theorem 4.31. I f  G in ~l ,~n~Pfi wil/1 I , :~3 ¢111¢/ ~7 ~1', tjlel! (~ i~ We also mention a result of ('hv(~lal and l~l'd~s. Theorein C [.4, Thcorcr..,. If. II' ( i  i.~ ~ /<-comwcled ~n4plm wi!h v ~ 3 ~md c~ ~< k, lhepl ( i  i,,~ 11~mlillotfi¢~PI. hon,,ly proved a common generalization of Theorems ![] and C. Existence (ff Dx .cycles and D x-paths 3 ~ 1 Theorem O [2, Theorem 2]. Let G he a k-connected graph with v ~ 3 such ~hat. for every, k + 1 mutually nonadjt~cent vertice~ uo, u~ . . . . .  u~ of G, k Y. atu,) >~(k + t)(v- ~). i =I) Then G is hamiitonian. In order to extend Theorems B, C and D to results on Ds-cycles for A > ! we need some additional definitions. As in [6], lwo subgraphs Ht al~d ;'~2 of a graph G are said to be close in G if they are disjoil~lt and there is an edge of G joining a vertex of H~ and one of H2, if no such edge exists in G, then HI and H:a, provided they are disjoint, are remote in G, Thus, if H'~ and H2 both consist of exactly erie vertex, Ht and H_~ are close (rem~te) ill* the corresponding verlices arc adjacetlt (nonadjacent). By t~ (G) (or just as) we denote the maximun~ number of mutually remote connected subgraphs of order ,~ of G. Thus t~t coincides with the independence number t~. The deg~v~e (~f a suhgraph H of G, denoted d(~(H) or d(H), is the number of vertices in V ( G ) -  V(H) adjacent Iv one or more vertices of H, In other words, considering vertices as subgraphs of order I, d(H) is lhe number of vertices of G close to H, If H centrists of a single vertex, then d(H) is just the degree of this vertex. The minimum degree of connected subgraphs of order A will be denoted 8~, so that d~ ~8,  If O is an oriented cycle or path in a graph and u and v are vertices on O. tlhen ( ) [u ,v]  and t)[t , ,u] denote, respectively, the segment of O from u to t~ and the reverse segment from t~ to 14. Furthermore, O(u,v]:~-=(~[u,v]~*{u}, 0[u,t~):=0[t~,t~]-*{v} a,~ld 0(u ,  v):-~ O[u, v]-..{u, v}. Three more defining relations are obtained hy revers- ing the arrows in the previous sentence, We are now ready to pr(we a generalization of Theorem C. Theorem 2. Let k ,rod ~ he positive integers such that either k ~" 2 or k ~ I and A ~ 2. If  G is a k-connected graph, other than a tree (in case k ~ 1), with (Y~ ~ k. then G is Ds-cyclic lhrool, By contrapcsition. Let G be a k-connected non-Ds-cyclic graph other than a tree, We will show that  as > k. Put t + I = min{i I G is D~-cyclic}, so theft t~A. Let C be a longest D~+t-cycle among all Df~l-cyctes C' of G for which to,(@- V(C')) is minimum. As in the proof of ~6, Theorem 3] one shows that C has length at least k + 1 ,  Fix an orientation on C. By assumption, C is a Dr+t-cycle, but not a /),-cycle of G. Hence G -  V(C) has a component He of order t. All vertices of 13 close to He are on C and, since G is k-connected and Iv(C)l~. k, we have that d(Ho)~,k. Let vt . . . . .  ~ok be k vertices of C close to He. For i -- 1 . . . . .  k, let u04 be a vertex of He adjacent to vd (for i ~ j, u0~ and uoj may coincide). Assume that ut . . . . .  vk occur on C in the order of their indices and let u~t be the immediate successor of v~ on C (i = 1 . . . . .  k). It will prove possible to 3~12 H.J. Veldman choose, for i = 1 . . . . .  k, a subgraph Hi of G satisfying the following require- ments: (i) Hi is connected and has order  t, (ii) H, CI C = t~[u~ ~, u~:~], where u~2 is a vertex of t~[u~ 1, v,) chosen in such a way that (iii) The  length of C'[u~t, u d  is minimum, i.e. if H is a connected subgraph of order t of G with H ~ C =  t~[u~, w], then ~ [u~ ,  u~z] is a subpath of t~[u~t, w]. Note that ~1 and u~ may coincide, in other words C'[u~ t, ~z]  may have length 0. If k = 1, then C may have length 3 and the existence of a subgraph H~ with the above propert ies is guaranteed only if t ~< 2. If k 1> 2, then, for 1 ~< i ~< k, (a) a subgraph ~ with ~he ment ioned propert ies exists, and (b) v~, ~ does not belong to C'[u~, u~2] (indices mod k). Assuming Ihe contrary to (a) or (b), consider the cycle c '  = V, Uo,~[ uo~, uo~, + ,]Uo., + , vi . t ~ [  vi + ~. v,], where P is a uo~uu.~ l-path within Ho (degenerate if uoi = u0.i+0. Ey assumption,  (~(v, v~÷~) is not contained in a component  of order  at least t of G - V(C') .  Since, moreover ,  tHo-V(C')l<t,  it follows that C '  is a Dr+t-cycle of G with t o t ( G -  V(C'))< to,(G-V(C)), contradicting the choice of C. Thus we have shown that, for 1 <~ i ~< k, a subgraph H~ satisfyi,~ the require-  ments (i), (ii) and (iii) indeed exists, provided t~<2 in case k = l. Following an analogous reasoning one proves that Ho and H~ are disjoint and. a fortiori, remote.  Next we prove by contradiction that, for 1 ~< i < ~ ~< k, the subgraphs Hi and Hi arc remote.  Assume that H~ and Hi are close or  non-disjoint. The:r. a u~2ui2-path P '  can be found such that ( i)  P'NC:= (~[u~2, w i ] U ( ' [ w  u ui2]., where wi and w~ are vertices of (~'[ui2, uil] and C[u~l, u~2], respectively, (2) no vertex of V ( P ' ) - V ( C )  is in /4o,  (3) the sum of the lengths of t~[u~2, w~] and (~[w~, ui2] is maximum,  i.e. no u~2ui.~-path satisfying (1) and (2) has more  vertices with C in common  than P'.  Now consider the cycle c"=  v, uoiP"[uo,, u,,,]uo,v~[TJ~, u,2]P'[u~:, u ~ ] ~ [ u i ~ ,  v,], where P" is a uo~uoi-path in /4 o. In Fig. I the cycle C" is indicated by arrows. Denote  by /~ and L~ the components  of G -  V(C") containing the vertices (if any) of (~[ui t, w,) and C[u~, wi), respectively. If L~ and L i would coincide, then a u~u~2-path satisfying (~) and (2) could be indicated having more  vertices with C in common tha~ P',  a contradiction with the choice of P'.  Thus ~ and Li are distinct, Moreover,  by the way Iti and H~ were chosen, b o t h / ~  a n d / ~  have order  less than t (other~vise (iii) would be violated). But then C" is a D ,+rcyc le  with t o , ( G -  V(C") )<  t o , ( G -  V(C)), contradict::ng the choice of C. Existence o[ D~-cycles and D~-paths I.I 0 " ~ t t t ' _  3 Fig .  I.  cl 313 Thus we have shown that the connected subgraphs /4o, Ht . . . . .  Hk of G of order t are mutually remote,  so that a, > k. Since % is easily seen to be a nonincreasing function of x, it follows that ax I> c~, > k. [ ]  For s ~ A ,  the graph K k v ( k + l ) K s  is non-Dx-cyclic and satisfies aA = k + l ,  showing that Theorem 2 is, in a sense, best possible. Theorem 2 can be improved to a generalization of Theorem D. Referring to the proof of Theorem 2, it can be shown that d ( H i ) + d ( H ~ ) ~ - v + k - A - k A  (O-<.i<]-<-k). Bondy [2] showed these inequalities to hold in case A = 1. The proof of the general case is completely analogous and hence omitted. Summing the above inequalities eventually yields Theorem 3. Let k and 3, be positive integers such that ei~'her k ~ 2 or k = 1 and A <~ 2. I f  G is a k-connected graph, other than a tree, such that, for every k + 1 mutually remote connected subgraphs Ho, Hi . . . . .  Ilk of order A of G, k d(Hi)>½(k + l ) ( v + k - h , - k ~ ) ,  i=O then G is D~-cyclic. 314 H.J. Vel4man Front Theorem 3 one easily deduces a generalization of  Theorem B: a k- connected graph with ~x >½(v + k - A - kA) is Dx-cyclic (k/> 2 or dk = 1 and X ~< 2). However, we can do better. Theorem 4, Let k and h be positive integers such that either k ~ 2 or k = 1 and A <~ 2. I f  G is a k-connected grapl~, other than a tree, wPh ~ ( v - ( k  + l )h  +k2) / (k  + l)  i f  h >-k, > ( ( v -  3`)/(h + 1) i f  3  ̀<~ k, then G is D~,-cyclic. l~'oof. By contraposition. Assume that G is k-connected and n o n - D : c y c l i c .  Set t +  ! =nfin{i [ G is D~-cyclic}, so that t x~3`. Let C be a D ,+ :cyc le  of G for which wf(G - V(C))  is minimum. We may assume C to have length at least k. Let H o be a component of G - V(C)  of order t and let v~ . . . . .  v,, be the vertices of C close to Ho, where m = d(Ho). Choose to each vi a subgraph Hi of G of order t as in the proof of Theorem 2 (i = 1 . . . . .  mL The choice of C then imphes, among other things, that the vertex sets V(Ho). V (HO . . . . .  V (H , , )  and {vl . . . . .  w,,} are mutually disjoint. Thus v >t (d(Ho) + 1)t + d(Ho), (l) or, equivalently, d(Ho) ~ (v - t)/(t + 1) and consequently &, ~ ( t , -  t)/(t + 1)+ t - h .  (2) Since G is k-connected, Ho has degree at least k, so (1) implies that v>~(k+ l)t +k.  (3) If 3, ~ k, then also t/> k. The inequality (3) is then equivalent to r - - t  v - ( k  + l)A + k  2 - - -  + t - A ~ ( 4 )  t + l  k + l  Combination of (2) and (4) proves the first part of the theorem. If 3  ̀~ k, then from (3) it follows that v>~(A + l)t'~ h. (5) Since t>~ 3 ,̀ the inequality (5) is satisfied if and only if v - t  v - - h  . . . .  4- t - 3` ~--:: ( 6 )  t + l  3`+1" The proof is now completed by combining (2) and (6). [ ]  Existence of D~,-cycles and Dx-paths 315 For h~>k, the collection { K k v ( k + l ) K , [ t ~ h }  consists of infinitely many k-connected non-Dx-cyclic graphs with ~ = ( v -  (k + 1)h + k2)/(k + 1). If h ~< k~ then {Kt v ( t  + 1)K~ [ t ~ k} is an infinite collection of k-connected non-Dx-cyclic graphs with ~x = (v-A)/(A + 1). Thus Theorem 4 is, in a sense, best possible. In view of Theorem 4, Theorem 3 might be improved to Conjecture 1. Let k and )t be positive integers satisfying either k >12 or k = l and h. <~ 2. If G is a k-connected graph, other than a tree, such that, for every k + 1 mutually remote connected subgraphs Ho, H1 . . . .  Hk of order X of G, k d(H~)> ~ v - ( k + l ) A + k 2  ifX>~k i=o t (k + 1 ) (v-h) / (h  + 1) if h ~ k ,  then G is D~-cyclic. If H is a subgraph of order k of a graph G and v is a 'vertex of H, then d(H)>~d(v) - k + 1. From this observation one easily deduces that the truth of Conjecture 1 (for h = k) would imply the truth of the following, which is a weaker version of a conjecture due to Bondy. Conjecture A (cf. [2, Conjecture 1]). Let G be a k-connected graph such that the degree-sum of every k + l  independent vertic,¢s is at least v + k ( k - 1 ) ,  where v ~> 3. Then there exists a cycle C of G such that G -  V(C) contains no path of length k -  1. In fact, Bondy conjectured that, under the condition of Conjecture A, every longest cycle C of G has the property that G--  V(C) contains no path of length k - 1 .  So far, the truth of Conjecture 1 has been established in the following case:~: (a) h = l  and k~>l (Theorem D), (b) A = 2  and k = i  [6, Theorem2] ,  (c) h = 2 and k = 2 [6, Cerollary 3.2]. Without giving it we mention that the proof of [6, Corollary 3.2] is easily extended to a proof of Conjecture 1 for k = 2 and h > 2. Theorem $. Let G be a 2-connected graph such ~hat the degree-sum of every three mutually remote connected subgraphs of order h >~ 2 is at least v - 3,X + 5. Then G is D~-cyclic. By Theorem 4, a 2-connected graph G has a D,x-cycle (h>12) if 8~ J~(v-3h +5). Under the assumption that GCY/'x the existence of a Dr-cycle can be proved if the weaker inequality 8~ ~>aX(v-3h + 3) is satisfied (the p~c, of is a 316 H.J. Veldman slight extension of the proof of  Theorem 4; instead of inequality (1) one demonstrates the inequality v 1> (m + !)t  + m + 2, where m = d(Ho), using the fact that delet:ion of the m vertices of C close to H0 does not create a gxaph with more than m components of order at least t). Thus, in particular, every 2-connected graph G satisfying G ~ X 2  and 6 2 ~ ] v - 1  is D2-cyclic, providing an extension of the foUowing consequence of a result of  Bigalke and Jung [1, Sate: 1]: a graph G with G¢ Xt and 6 ~>]v has a D2-cycle. The latter result, in turn, easily implies the following, due to Nash-Williams [5, Lemma 4]: if G is a 2-connected graph and /5/> ma,~(a, .~(v + 2)), then G is hamiltonian. References [I] A. Bigalke and H.A. Jung, l lber Hamiltonsche Kreise und unabh~ingige Ecken in Graphen, Mcmatsh. Math. 88 (1979) 195-210. [2] J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada (1980). [3] J.A. Bondy and U.S.R. Murty. Graph Theory with Applications (Macmillan, London and Elsevier, blew York, 1976). [4] V. Chvfital and P. Erd6s, A note on hamiltonian circuits, Discrete Math. 2 (197~) 111-113. [5] ('. St. J.A. Nash-Williams, Edge-disjoint hamiltonian circuits ii: graphs with vertices of large valency, in: L. Mirsky, ed., Studies in Pure Mathematics (Academic Press, London, 1971) 157--183. [6] !H.J, Veldman, Existence of dominating cycles and paths, Discrete Math. 43 (19~31 281-296. 

Existence of Dλ-cycles and Dλ-paths

AbstractA cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Elsevier - Publisher Connector 

Discrete Mathematics 44 (1983) 309-3i6 309 
North-Holland Publishing Company 
EX ISTENCE OF  D. , , - ,CYCLES AND Dx-PATHS 
H.J. VELDMAN 
Department ofA~plied Mathematics, Twente University of Technology, Enschede, the Nether- 
lands 
Received 4 March 1982 
Revised 28 May 1982 
A cycle C of a graph G is called a D A-cycle if every component of G - V(C) has order less 
than A. A Dx-path is defined analogously. In particular, aDl-cycle is a hamiltonian cycle and a 
Dl-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for 
graphs to have a Dx-cycle or Dx-path. The results are generalizations of theorems in 
hamiltonian graph theory. Extensions of notions uch as vertey degree and adjacency of vertices 
to subgraphs of order greater than 1 arise in a natural way 
1. Introduction 
We employ the terminology of Bondy and Murty [3] and consider only simple 
graphs. 
In [2], Bondy stated a sufficient condition for a graph G to have a cycle C such 
that G - V(C)  contains no Kk. For  k = 1, it coincides with Ore 's  condition for the 
existence of a hamiltonian cycle. Here we introduce another kind of generalized 
hamiltonian cycle. A cycle C of a graph G is a D~-cycle if all components of 
G - V(C)  have order less than A. Alternatively, C is a Dx-cycle of G if and only if 
every connected subgraph of order A of G has at least one vertex with C in 
common. Thus a Dx-cycle dominates all connected subgraphs of order ~. Analog- 
ously, a path P of G is a D~-path if every component of G - V(P) has order less 
than A. Graphs containing a Dx-cycle (Dx-path) will be called Dx-cyclic (Dx- 
traceable). A Dl-cycle (Dl-path) is the same as a hamiltonian cycle (hamiltonian 
path). D2-cycles were studied in [6]. 
In subsequent sections, existence theorems for Dx-cycles are proved. In [6], 
most of them were already proved for A = 2. W~ will henceforth refrain from 
referring to these special cases, unless this is essential. Parallel results on Dx-paths 
can be obtained, using the following obvious lemma. 
Lemma 1. A graph G is Dx-traceable i[ and only i[ GvK~ is Dx-cyciic. 
The theorems derived are generalizations of known result.,; in hamiltonian graph 
theory. A corresponding remark can be made about the proof techniques used. 
Some of the results in Section 3 are closely related to Bondy's work [2]. 
0012-365XI8310000-0000/$03.00 © 1983 North-Hol land 
310 H,J, Veldmrn 
Exte~,,sions to subgraphs of order greater than 1 of concepts such ab adjacency 
of vertices, independence number and vertex degree arise in correspondence with 
the generalization of hamiltonim~ cycles to D,~-cycles, 
2. A necessary condition in ~erms of cu~ sets 
+I'0 stt~,l't with, we generalize a necessary condition for the existence of ~.~. 
hamiltonia,~ ,.'vole, 
~heorem A 13, Theorem 4,21. Ii" (~ ,~r~l,.Iv ( ;  i,~ l~,~M;i~mi,,,,, flyers, ./'or eeery 
¢'~((; ,~"<" I,% 
I)~.',,~ote bv ~o~ ( ( ; l  the ntlnll'~cr of .--onlpotletlts ,,)f ( i  ~1 order i~t lei,tsl A, Th@v,t'etll 
,,\ is then a special case (A' I] oF 
Theorem I. I f  ~ ,~r~l)le (;  i.~ !)~-cyclic, Ilwn, for ~'rerv ~l~meml)ly pn~),'r .~d~.wl S of 
~.'((;L 
Th,; pr+.~ol, b¢in.e, an c~tsy extension of the proof of I6, Theorem ~I. is omiltcd, 
l,'or f~lltlt't~, refer~.,n~.'e we t|ellote bv ;)C,~ the ~.'hlss oP' graphs t~.~ salisfyil|g the 
nccc,,,s~ry condition of Theorem I, Thus ( i  is in ,/~.~ i1!', for some uo~ ~ply  proper 
.,,tll'~s~,,t ~ o!' ~'((;). ,.,,~((i ,~'I~-I,~I , 
3. Sufficient conditions involving subllraph dellrees 
We now turn our attentinn h~ sutlicicnt conditions for rite cxislcnce of 
l)~-¢yclcx, (Inc ~l' the earliest res:dts ill hanliltoni;m graph theory to be 
gcncralitcd hcl"c is due !o l)ira¢. 
' rh~wem B j3, Theorem 4.31. I f  G in ~l ,~n~Pfi wil/1 I,:~3 ¢111¢/ ~7 ~1', tjlel! (~ i~ 
We also mention a result of ('hv(~lal and l~l'd~s. 
Theorein C [.4, Thcorcr..,. If. II' ( i  i.~ ~ /<-comwcled ~n4plm wi!h v ~ 3 ~md c~ ~< k, 
lhepl ( i  i,,~ 11~mlillotfi¢~PI. 
hon,,ly proved a common generalization of Theorems ![] and C. 
Existence (ff Dx .cycles and D x-paths 3 ~ 1 
Theorem O [2, Theorem 2]. Let G he a k-connected graph with v ~ 3 such ~hat. 
for every, k + 1 mutually nonadjt~cent vertice~ uo, u~ . . . . .  u~ of G, 
k 
Y. atu,) >~(k + t)(v- ~). 
i =I) 
Then G is hamiitonian. 
In order to extend Theorems B, C and D to results on Ds-cycles for A > ! we 
need some additional definitions. As in [6], lwo subgraphs Ht al~d ;'~2 of a graph 
G are said to be close in G if they are disjoil~lt and there is an edge of G joining a 
vertex of H~ and one of H2, if no such edge exists in G, then HI and H:a, provided 
they are disjoint, are remote in G, Thus, if H'~ and H2 both consist of exactly erie 
vertex, Ht and H_~ are close (rem~te) ill* the corresponding verlices arc adjacetlt 
(nonadjacent). By t~ (G) (or just as) we denote the maximun~ number of mutually 
remote connected subgraphs of order ,~ of G. Thus t~t coincides with the 
independence number t~. The deg~v~e (~f a suhgraph H of G, denoted d(~(H) or 
d(H), is the number of vertices in V(G) -  V(H) adjacent Iv one or more vertices 
of H, In other words, considering vertices as subgraphs of order I, d(H) is lhe 
number of vertices of G close to H, If H centrists of a single vertex, then d(H) is 
just the degree of this vertex. The minimum degree of connected subgraphs of 
order A will be denoted 8~, so that d~ ~8, If O is an oriented cycle or path in a 
graph and u and v are vertices on O. tlhen ()[u,v] and t)[t,,u] denote, 
respectively, the segment of O from u to t~ and the reverse segment from 
t~ to 14. Furthermore, O(u,v]:~-=(~[u,v]~*{u}, 0[u,t~):=0[t~,t~]-*{v} a,~ld 
0(u, v):-~ O[u, v]-..{u, v}. Three more defining relations are obtained hy revers- 
ing the arrows in the previous entence, 
We are now ready to pr(we a generalization of Theorem C. 
Theorem 2. Let k ,rod ~ he positive integers such that either k ~" 2 or k ~ I and 
A ~ 2. If G is a k-connected graph, other than a tree (in case k ~ 1), with (Y~ ~ k. 
then G is Ds-cyclic 
lhrool, By contrapcsition. Let G be a k-connected non-Ds-cyclic graph other 
than a tree, We will show that as > k. Put t + I = min{i I G is D~-cyclic}, so theft 
t~A. Let C be a longest D~+t-cycle among all Df~l-cyctes C' of G for which 
to,(@- V(C')) is minimum. As in the proof of ~6, Theorem 3] one shows that C 
has length at least k+1,  Fix an orientation on C. By assumption, C is a 
Dr+t-cycle, but not a /),-cycle of G. Hence G-  V(C) has a component He of 
order t. All vertices of 13 close to He are on C and, since G is k-connected and 
Iv(C)l~. k, we have that d(Ho)~,k. Let vt . . . . .  ~ok be k vertices of C close to He. 
For i -- 1 . . . . .  k, let u04 be a vertex of He adjacent to vd (for i ~ j, u0~ and uoj may 
coincide). Assume that ut . . . . .  vk occur on C in the order of their indices and let 
u~t be the immediate successor of v~ on C (i = 1 . . . . .  k). It will prove possible to 
3~12 H.J. Veldman 
choose, for i = 1 . . . . .  k, a subgraph Hi of G satisfying the following require- 
ments: 
(i) Hi is connected and has order t, 
(ii) H, CI C = t~[u~ , u~:~], where u~2 is a vertex of t~[u~ 1, v,) chosen in such a way 
that 
(iii) The length of C'[u~t, ud  is minimum, i.e. if H is a connected subgraph of 
order t of G with H~C= t~[u~, w], then ~[u~, u~z] is a subpath of t~[u~t, w]. 
Note that ~1 and u~ may coincide, in other words C'[u~ t, ~z] may have length 0. 
If k = 1, then C may have length 3 and the existence of a subgraph H~ with the 
above properties is guaranteed only if t ~< 2. 
If k 1> 2, then, for 1 ~< i ~< k, 
(a) a subgraph ~ with ~he mentioned properties exists, and 
(b) v~, ~ does not belong to C'[u~, u~2] (indices mod k). 
Assuming Ihe contrary to (a) or (b), consider the cycle 
c '  = V, Uo,~[ uo~, uo~, + ,]Uo., + , vi . t ~[  vi + ~. v,], 
where P is a uo~uu.~ l-path within Ho (degenerate if uoi = u0.i+0. Ey assumption, 
(~(v, v~÷~) is not contained in a component of order at least t of G - V(C'). Since, 
moreover, tHo-V(C')l<t, it follows that C' is a Dr+t-cycle of G with to t (G-  
V(C'))< to,(G-V(C)), contradicting the choice of C. 
Thus we have shown that, for 1 <~ i ~< k, a subgraph H~ satisfyi,~ the require- 
ments (i), (ii) and (iii) indeed exists, provided t~<2 in case k = l. Following an 
analogous reasoning one proves that Ho and H~ are disjoint and. a fortiori, 
remote. 
Next we prove by contradiction that, for 1 ~< i < ~ ~< k, the subgraphs Hi and Hi 
arc remote. Assume that H~ and Hi are close or non-disjoint. The:r. a u~2ui2-path 
P' can be found such that 
(i) P'NC:= (~[u~2, w i ]U( ' [w u ui2]., where wi and w~ are vertices of (~'[ui2, uil] 
and C[u~l, u~2], respectively, 
(2) no vertex of V (P ' ) -V (C)  is in/4o, 
(3) the sum of the lengths of t~[u~2, w~] and (~[w~, ui2] is maximum, i.e. no 
u~2ui.~-path satisfying (1) and (2) has more vertices with C in common than P'. 
Now consider the cycle 
c"= v, uoiP"[uo,, u,,,]uo,v~[TJ~, u,2]P'[u~:, u~]~[u i~,  v,], 
where P" is a uo~uoi-path in /4 o. In Fig. I the cycle C" is indicated by arrows. 
Denote by /~ and L~ the components of G-  V(C") containing the vertices (if 
any) of (~[ui t, w,) and C[u~, wi), respectively. If L~ and L i would coincide, then a 
u~u~2-path satisfying (~) and (2) could be indicated having more vertices with C in 
common tha~ P', a contradiction with the choice of P'. Thus ~ and Li are 
distinct, Moreover, by the way Iti and H~ were chosen, both/~ and/~ have order 
less than t (other~vise (iii) would be violated). But then C" is a D,+rcycle with 
to , (G-  V(C"))< to , (G-  V(C)), contradict::ng the choice of C. 
Existence o[D~-cycles and D~-paths 
I.I 0 " 
~ t t t ' _  
3 
Fig.  I. 
cl 
313 
Thus we have shown that the connected subgraphs /4o, Ht . . . . .  Hk of G of 
order t are mutually remote, so that a, > k. Since % is easily seen to be a 
nonincreasing function of x, it follows that ax I> c~, > k. []  
For s~A,  the graph Kkv(k+l )Ks  is non-Dx-cyclic and satisfies aA =k+l ,  
showing that Theorem 2 is, in a sense, best possible. 
Theorem 2 can be improved to a generalization of Theorem D. Referring to the 
proof of Theorem 2, it can be shown that 
d(H i )+d(H~)~-v+k-A-kA  (O-<.i<]-<-k). 
Bondy [2] showed these inequalities to hold in case A = 1. The proof of the 
general case is completely analogous and hence omitted. Summing the above 
inequalities eventually ields 
Theorem 3. Let k and 3, be positive integers such that ei~'her k ~ 2 or k = 1 and 
A <~ 2. If G is a k-connected graph, other than a tree, such that, for every k + 1 
mutually remote connected subgraphs Ho, Hi . . . . .  Ilk of order A of G, 
k 
d(Hi)>½(k + l ) (v+k-h , -k~) ,  
i=O 
then G is D~-cyclic. 
314 H.J. Vel4man 
Front Theorem 3 one easily deduces a generalization of Theorem B: a k- 
connected graph with ~x >½(v + k - A - kA) is Dx-cyclic (k/> 2 or dk = 1 and X ~< 2). 
However, we can do better. 
Theorem 4, Let k and h be positive integers such that either k ~ 2 or k = 1 and 
A <~ 2. I f  G is a k-connected grapl~, other than a tree, wPh 
~ (v - (k  + l)h +k2)/(k + l) if h >-k, 
>((v -  3`)/(h + 1) i f  3  `<~ k, 
then G is D~,-cyclic. 
l~'oof. By contraposition. Assume that G is k-connected and non-D:cyc l ic .  Set 
t+ ! =nfin{i [ G is D~-cyclic}, so that t x~3`. Let C be a D,+:cycle of G for which 
wf(G - V(C)) is minimum. We may assume C to have length at least k. Let H o be 
a component of G - V(C) of order t and let v~ . . . . .  v,, be the vertices of C close 
to Ho, where m = d(Ho). Choose to each vi a subgraph Hi of G of order t as in 
the proof of Theorem 2 (i = 1 . . . . .  mL The choice of C then imphes, among 
other things, that the vertex sets V(Ho). V(HO . . . . .  V(H, , )  and {vl . . . . .  w,,} are 
mutually disjoint. Thus 
v >t (d(Ho) + 1)t + d(Ho), (l) 
or, equivalently, 
d(Ho) ~ (v - t)/(t + 1) 
and consequently 
&, ~(t , -  t)/(t + 1)+ t -h .  (2) 
Since G is k-connected, Ho has degree at least k, so (1) implies that 
v>~(k+ l)t +k. (3) 
If 3, ~ k, then also t/> k. The inequality (3) is then equivalent o 
r - - t  v - (k  + l)A +k  2 
- - -  + t - A ~ (4 )  
t+ l  k+l  
Combination of (2) and (4) proves the first part of the theorem. 
If 3  `~ k, then from (3) it follows that 
v>~(A + l)t'~ h. (5) 
Since t>~ 3 ,` the inequality (5) is satisfied if and only if 
v- t  v - -h  
. . . .  4- t - 3` ~--:: (6 )  
t+ l  3`+1" 
The proof is now completed by combining (2) and (6). []  
Existence ofD~,-cycles and Dx-paths 315 
For h~>k, the collection {Kkv(k+l )K , [ t~h} consists of infinitely many 
k-connected non-Dx-cyclic graphs with ~ = (v -  (k + 1)h + k2)/(k + 1). If h ~< k~ 
then {Kt v(t  + 1)K~ [ t ~ k} is an infinite collection of k-connected non-Dx-cyclic 
graphs with ~x = (v-A)/(A + 1). Thus Theorem 4 is, in a sense, best possible. 
In view of Theorem 4, Theorem 3 might be improved to 
Conjecture 1. Let k and )t be positive integers atisfying either k >12 or k = l and 
h. <~ 2. If G is a k-connected graph, other than a tree, such that, for every k + 1 
mutually remote connected subgraphs Ho, H1 . . . .  Hk of order X of G, 
k d(H~)> ~ v - (k+l )A+k2 ifX>~k 
i=o t (k + 1)(v-h)/(h + 1) if h~k,  
then G is D~-cyclic. 
If H is a subgraph of order k of a graph G and v is a 'vertex of H, then 
d(H)>~d(v) - k + 1. From this observation one easily deduces that the truth of 
Conjecture 1 (for h = k) would imply the truth of the following, which is a weaker 
version of a conjecture due to Bondy. 
Conjecture A (cf. [2, Conjecture 1]). Let G be a k-connected graph such that the 
degree-sum of every k+l  independent vertic,¢s is at least v+k(k -1) ,  where 
v ~> 3. Then there exists a cycle C of G such that G-  V(C) contains no path of 
length k -  1. 
In fact, Bondy conjectured that, under the condition of Conjecture A, every 
longest cycle C of G has the property that G-- V(C) contains no path of length 
k -1 .  
So far, the truth of Conjecture 1 has been established in the following case:~: 
(a) h=l  and k~>l (Theorem D), 
(b) A=2 and k=i  [6, Theorem2], 
(c) h = 2 and k = 2 [6, Cerollary 3.2]. 
Without giving it we mention that the proof of [6, Corollary 3.2] is easily 
extended to a proof of Conjecture 1 for k = 2 and h > 2. 
Theorem $. Let G be a 2-connected graph such ~hat he degree-sum of every three 
mutually remote connected subgraphs of order h >~ 2 is at least v - 3,X + 5. Then G is 
D~-cyclic. 
By Theorem 4, a 2-connected graph G has a D,x-cycle (h>12) if 8~ 
J~(v-3h +5). Under the assumption that GCY/'x the existence of a Dr-cycle can 
be proved if the weaker inequality 8~ ~>aX(v-3h +3) is satisfied (the p~c, of is a 
316 H.J. Veldman 
slight extension of the proof of Theorem 4; instead of inequality (1) one 
demonstrates the inequality v 1> (m + !)t + m + 2, where m = d(Ho), using the fact 
that delet:ion of the m vertices of C close to H0 does not create a gxaph with more 
than m components of order at least t). Thus, in particular, every 2-connected 
graph G satisfying G~X2 and 62~]v-1  is D2-cyclic, providing an extension of 
the foUowing consequence of a result of Bigalke and Jung [1, Sate: 1]: a graph G 
with G¢ Xt and 6 ~>]v has a D2-cycle. The latter result, in turn, easily implies the 
following, due to Nash-Williams [5, Lemma 4]: if G is a 2-connected graph and 
/5/> ma,~(a, .~(v + 2)), then G is hamiltonian. 
References 
[I] A. Bigalke and H.A. Jung, llber Hamiltonsche Kreise und unabh~ingige Ecken in Graphen, 
Mcmatsh. Math. 88 (1979) 195-210. 
[2] J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, 
Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, 
Waterloo, Ontario, Canada (1980). 
[3] J.A. Bondy and U.S.R. Murty. Graph Theory with Applications (Macmillan, London and Elsevier, 
blew York, 1976). 
[4] V. Chvfital and P. Erd6s, A note on hamiltonian circuits, Discrete Math. 2 (197~) 111-113. 
[5] ('. St. J.A. Nash-Williams, Edge-disjoint hamiltonian circuits ii: graphs with vertices of large 
valency, in: L. Mirsky, ed., Studies in Pure Mathematics (Academic Press, London, 1971) 
157--183. 
[6] !H.J, Veldman, Existence of dominating cycles and paths, Discrete Math. 43 (19~31 281-296. 


Existence of Dλ-cycles and Dλ-paths

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