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Forbidden subgraphs and Hamiltonian properties of graphs
Discrete Mathematics42 (1982) 1 8 9 - 1 9 6 : North-HollandPublishingCompany
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,:
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189
FORB~DEN S~GIIAPFtS AND EIAMILTONIAN PROPERTIES OF GRAPHS Ronaid J. G O U L D *
Emory University, Atlanta, GA 30322, USA Michael S. JACOBSON**
university of Louisville, Louisville, KY 40292, USA Received 24 November 1980 R¢vised 14 September 1981 and 21 December 1981 ~¢adous sufficientconditionsare giveii,in terms of forbiddensubgraphs, that implya graph is ei~hc~homogeneouslytraceable, hamiltoni~nor pancyclic. We consider only finite undirected graphs without loops or multiple edges. No~atie, or terms not defined here can be found in [1]. Let G be a graph and Iet S ~ V(G). The subgmph (S) induced by S is the graph with vertex set S and who~t~ edge set consists of those edges of G incident with two vertices of S. The distence d(u, v) between vertices u and v in a connected graph G is the minimum number of edges in a u - v path. The diameter of a graph G is max .... yea) d(u, v). A graph is hamiltonian (traceable) if it has a cycle (path) contaipSng all its vertices. A pancyclic graph of order p conlains a cycle of length l for eac~ ,~ (3 <~l ~
TheoremA
[4]. If G is a 2-connected graph that contains no induced sr~bgraph isomorphic to KL3 or Z~, then G is hamiltonian. We note that the proof of Theorem A actually shows that either G is a cycle or * Research supported by a grant from Emory University. ** Research supported by a grant from the Universityof Louisville.
0012-365X/82/0000-0000/$02.75 © 1982 North-Holland
199
R.J.
Gould, M.S.
Ja~bson
G is pancyclic. We now show that slightly more general conditions yield the same result.
Theorem 1. I[ ,G is a 2-connected gral:h that contains no induced subgraphs isomorphic to Kr 3 or Z~, then G is a cycle or pancyclic°
Proo|. The result is trivial if ]V(G)I<~4. So ,,mppose G satisfies the hypothesis, is not a cycle, and ilas order at least 5. Let C: vov~ • • • vk-lVo ( k >I 2) be an arbitrary cycle of length k. We show that if V ( G ) ¢ V ( C ) , then a (k + 1)-cycle can be found. Since G is not a cycle and contains no induozd KL3, then it must contain a 3-cycle. Since G is not connected there e:tists Xoe V ( G ) - V ( C ) such that Xo is adjacent to a vertex of C. Without loss of generality we may assume Xo is adjacent to 01 (or else relabel C), There is no induced K~.3 in G so the edge roY2 is in G or a (k + l)-cycle would be produced. Since G is 2-connected, there exists an x 0 - v2 path P, not conlaining vl. Consider the subpath P': XoXl • • • xlvi where vj is the first vertex of C on P. Case 1. Supp~se vi : ~2 (or by symmetry vj = Vo). The graph ({VO, 1)1, D2, XO,X1})~-Z2 unless one of the edges XlVo, x~vl or xtv2 is present. Consider x~v2 (and note a similar argument holds for xlvo). If vk_l ='- v2, then xoxlv2v~xo is a cycle of length 4. If vk-~ ~ v2, then ({v~, v2, Va, xz}) implies that v~ v.a ~ E ( G ) . If 1~_~ = 03, then vovlxox~v2vo is a 5-cycle. If Vk-~ ~ V3, we consider ({v~, vl, v3, xo}). If xov3 is an edge of G, then v o v 2 x l x o v 3 V 4 " " v k - l V o is a ( k + l ) - c y c l e . Hence we see that VoV~ is an edge of G. The graph ({ vo, v2, v3, x t xo})----Z~_unless one of the edges XoV3, x~ vo or J:~v3 is in G. But the inclusion of any of these edges produces a (k + 1)-cycle. If xl v~ ~ E ( G ) , we may consider the path P": x~x2" • • x~v2 and repeat the above argument. Eventually, either the previous possibility results or a (k + 1)-cycle is formed. Case 2. Suppose 3 ~ f ~< k - 1. Since G contains no induced Z2, it is easily seen that for some i (0<~ i ~< l), v~x~ and x~vj are edges of G. Observe that v~# vi_~ and vk I ¢ v~ or a (k + 1)-cycle can be found. Further, oi_iv~+t ~ E ( G ) since G contains no induced KI.3. Consider ({v~,, vl, v2, x~, vi}). It follows tha~ either ViVo~ E ( G ) in wh~.ch case I)oVk_ t . . . .
V~+~l)i_ l ..-
VlXiViV 0
is a (k + 1)-cycle or v i v 2 ~ E ( G ) , and VoVk_t
...
vi+ivi_
~ ...
v2vixiv~v
o
is a (k + ][)-cycle. Thus viv~ ~ E ( G ) and ({v~, xi, v i, v2, v3})----Z2 unless at least one of x~v2, x, vs, viv2, t~iv3 or v~v3 is an edge of G. The first three yield immediate (k + l)-cycles. If viv3a E ( G ) , then port,_
~ • , . VI+~I)i_ ~ ,..
is a (k + l)-cycle.
I)3I)iXiD~I)2V 0
191
Forbiddensubgraphsand harniltonianproperties
Finally, ff v l v 3 e E ( G ) , then ({Co,el, v2, vj, v~-I})---Z2 nless one of VoVj, v2vj, VoOi-l, v2vi-1 or v1%-1 is ~n G. But each edge produce~ a (k + 1)-cycle and the result follows. [] Remark lo We note that the hypothesis of Theorem I does not imply that the graph is panconnected. Consider K, (n t> 3) with vertices x , x 2 , . . . , x,. Let G be that graph obtained by subdividing the edge xtx2 and name the new vertex x. Thi~ graph does not contain an induced Kt, a or Z2 yet there is no x - x 1 path of length 2. Furth~'rmore, we cannot omit either of the induced subgtaphs from the hypothesis. Fig. l(a) shows a nonhamiltonian graph with no induced Z2. It is constructed by taking two copies" of C2~÷~ ( n > 1) and joining corresponding vertice~ in each copy by a path of length 2. Fig. l(b) shows a nonhamiltonian graph w~th no induced KL3. It i~ constructed by taking two copies of K2n.~ (n > 1) and ioinlng corresponding vertices in each copy by an edge and a path of length 2.
(a)
Contains no
Z2 .
(b)
Contains
no
Kl,, ~ .
F i g . 1.
The following was shown in [3]. Theorem B. If G is a 2-connected graph that contains no induced subgraph isomorphic to Kt.3 or F (see Fig~ 2(a)), then G is hamiltonian. We note that the condition~ of this theorem are not enough to imply that the graph be pancyclic; for example the graph of Fig. 2(b) is 2-connected, contains no induced subgraph isomorphic to Kt,3 or F and is not pancyclic. However it is easily shown that the hypothesis of Theorem 1 implies the hypothesis of Theorem B. The proof is routine and not included. Proposition 2. If G is 2-connected and contains no induced subgraph isomorphic to KI,a or Z2, then G contains no induced subgraphs isomo~hic to F.
R.I. Gould, M.S. lacobson
192
T]~.~ graph
F .
(b)
A no'e.pancycllc g r a p h .
Fig. 2.
The graph in F~g. l(b) shows that Z~ cannot be replaced by Z3 in Theorem 1.. In The~rem 3 we raodify the set of forbidden subgraphs to include Z3, but the c~nclusion i~ weaker than ~hat of Theorem i. Let B be that graph obtained by k~entifying a vertex in two distinct copies of K 3. "l'heo~rem 3. I[ G is a 2..connected graph that contains no induced subgraph isomorphic ~c. K;,:~, B, or Z~, then G is hamiltonian. Proof. Suppo,,;e G satisfies the hypothesis and is not hamiltonian. Choose a cycle C: t~,,t,~ " " ~k~:o (k > 2) of maximum length in G. Let xo~ V ( G ) - V ( C ) such ~hat x,, is adiaccnt w,, a vertex of C~ Without loss of generality suppose Xo is adjacent to tq. Siricc K~.3 is not an induced subgraph of G, V o v 2 e E ( G ) or a longer cycle would be present. Since G is 2-cop~nected, there must exist an x o - '02 path P that does r~ot contain t~. Suppose ~,/ is the first vertex of P on C. It is clear that t', , t~j. L~ E ( G ) and that j ~ 0, 2, 3, 4, k -- 1, or k, for otherwise C would not be of maximur,n length. Since G contains no induced Z3, we cant find a v ~ - v j path P', that is :~ ~ubpath of P. and is disjoint from V ( C ) - { v ~ , vj}. Further, the length of P' is a~ mc~st 3. Suppose the ~ength of P' is 3. that is, suppose P' is the path VtXoX~Vj. By considering ({v,~. v~, u:. x~, x~. t~i}),one can readily establish that v~v/is an edge of G. The subgraph ({t~, v2, J"o, t:/})~- Kt.3 unless v,_t) is an edge of G. But now the cycle l)~l~t~i Xi,Xll.~il)2L13
" " " tli
1U/~l/)kt~O
has length lenge~ than C which is a contradiction. Thcrefore we may assume that the length of P' is 2, that is, suppose P' is the path t, lx,¢o/. We tirst consider the graph ({vo, v~, v:,Xo, vt, v/.~}). We need only cow,sider whether ~ot), v~vj, vav i, t,,~e/, t, v~ vi~.~ or v2vi. ~ are edges of G. If any one of v,~v~, z~_v~, v~v~÷~, or v ~ v ~ is an edge of G, a cycle longer than C is immediately produced. If v~ v~ ~ E ( G ) , the~ ({v0, v~, vz, xo, v~})-~B, unless G contains an edge previously considered. Thus v~vj.~ is an edge of G and we next consider ({ro. t~, t,i,., t,j,z}), If v~v~.:~ is an edge of G, then either ({q-t, Xo, vi, Vi+2})~KI,3 or a Iongt~r cycle exists. The edge vov~ has already been handled; hence we conclude that t~ov/., is an edge of G. Now the graph ({v6, vl.2, vi+ ~, vi, v~-a})~-B
Forbidden subgraphsand hamiltonian properaes
193
unless an edge already analyzed exists in G. Thus, all cases produce a cycle longer than C, and hence we conclude that G is hamiltonian Q Remark 2. The hypothesis of Theorem 3 does not imply that of Theorem B (see Fig. 3). We also note that the hypothesis of "l~eorem 3 does not imply the graph is pancyclic (see Fig. 2(b)).
Fig. 3. A graph with no induced K1,3,Z3 or B that containsan induced F. Theorero C [5]. If G is a 2-connected graph o/diameter at most 2 that contains no indt:ce,~ :ubgraph isomorphic to Kl,3, then G is hamiltonian. We rJote that 2-connected nontraceable graphs of diameter 3 with no induced K:,:~ can be found (see Fig. 4). However, we may modify the set of forbidden subgra~hs to produce homogeneously traceable graphs of diameter at most 3.
K3n (n_>3)
Fig. 4. A graph with diameter 3 and no induced K1,3that is not traceable.
194
1~.3. Gould, M.S. 1acobson
Theorem 4. r[ G is a 2-cormected graph o f d i a m e t e r at m o s t 3 a n d G contains no i n d u c e d s.ub,graph isomorphic to Kt,3 or B, then G is h o m o g e n e o u s l y traceable. Proof. By Theorem C we need only cor~sider graphs of diameter 3. So suppose C satisfies the hypothesis of the Theorem but is not homogeneously traceable. Thus there exists a vertex vo that is not the initial vertex of a hamiltonian path in G. Let P: v o v ~ ' " v , be a longest path with initial vertex vo. Thus there exists x e V ( G ) - V ( P ) such that x is adjacent to a vertex v~ of P. Furthermore, we may as~urne i < n or a longer path would be evident and we may assume 0 < i or else v,, would be a cutvertex. Observe that v~..~v~+t ~ E ( G ) . Hence i < n - 1 . Since 2 ~- d(x, v,,) < 3 we consider two cases. C a s e 1. Suppose d ( x , vn) :: 2. Since v,, must be adjacent only to vertices of P, the vertex intermediate to x and v. on the distance path must lie on P, If XVoV. is the path of length 2, then ({x, Vo, vt, v . } ) ¢ K~,s implies that v~v. is an edge of O. It follows that the path voxv~v~+~ • • • v . v ~ v : • " • v~_t is longer than P and has initial ~:crtcx v~,. Thus we may assume xv~v. (0 < i < n) is the distance path, or we would merely change our choice of vi above. Now VoVj • • • v~-~vHvi+2 • " • l)ntliX is a path longer than P. Case 2. Suppose d(x, v . ) = 3. Let xv~qv, be the x - v . distance path. Clearly t~i. t~t ,5/9.
';ubcase 2a. Suppose j = 0. Since ({v0, v~, v~, 1)n})~K1,3 at least one of viva, v,v,, or v~v,, is an edge of G. Except for v~v. longer paths are easily established.
So suppose v~v. ~ E ( G ) and consider {{vo, v~, vi+~,x}). The edges xv~+~ and xvo yield longer paths easily; while if vovi+t ~ E ( G ) the path t'oU,, t l - ' i ~ 2 . . . . 1 . l n O l l ) 2 ' ' " 1)i_t/)iX is a longer path. 5 u b c a s e 2b. Suppose 0 < i < i - 1. By considering ({v i, vi+~, vi, v.}) and ({t'j,t~ j , v . v , , } ) we must have v~vj_~ and vi÷Lv, as edges of G. Since ({v,, t'~ ~. t',, q , ~ . v . } ) ~ B at least one of r~,i_~vj?~, vi_~v., v~vi+~ and v~v. is an edge of G. However. a longer path results in each case. S u b c a s e 2c. Soppose j = i - 1. The path vov~ " • • v~_tv.v._~ " • • vix is a longer path. S~bcase 2d. Suppose j : = i + l ~ n - - 1 (j=n-1 lis Subcase 2f). Since ({r,.vi,l,t,~e,v.})~K~,3 we can conclude that v.~'~i+z~E(G). But since ({v, t, v~, t , ~ , v~.., v , , } ) ~ B at least one of vi_~vi+~, vi-~v., viv~.,z or viv. is in G. Again a longer path is easily established in each case. S u b c a s e 2e. Suppose i + 1 < j < n - 1. The argument for this case is analogous to that of Subcase 2b. .~ubcase 2f. Suppose j = n - 1 . With this being the final possibility we may assume for each longest path P of length n with initial vertex Vo and final vertex t',, and arly vertex x not on P, that d(x, v . ) = 3 and x v ~ v . _ t v , is a dista.:ce path where 1 ~ i ~< n - 2. Since G is 2-connected v. is adjacent to some vertex of P, say Vk. Clearly k5e i - 1 or i. If 0 < k < i - 1 and since ({v~, v., v~+l, th:_l})~ K:l.3, then v,,t,~ t, v.vt,, ~ or v~_ ~,Vk+~ is in G. Longer paths are immediate for the first two; so
Forbidden subgraphsand hamiltonianproperties
195 ~4
suppose Vk+~Vk-I is an edge of G. Then 0.: VoVl " " • Vk- tOk+1 " " " VnVk is a longest path with initial vertex Vo. Further, x~ V(O) so the distance from x to on is 2. But this contradicts t h e fact that d(x, v,,) = 3. A similar argumen'~ applies when i < k < n - 1. Hence- we may additionally assume that the last vertex of every longest path with initial vertex vo is adjacent to Vo. Consider (~'.
I)OV 1 • . . V/_lVi+
l • . . ~)~_lV[X.
Now O is a longest path with initial vertex vo and end vertex x so that xvo~ E(G). The~ ({vo, x, vt, vn})~Kl.3 unless one oY xv~, xv. or v~v, is an edge of G. In any case a new path longer than O (or P) is apparent. [] Since aomogeneously traceable nonhamiltonian graphs have no vertices adjacent to t~/o or more vertices of degree 2, the following is immediate. C o m f i a ~ 5. ~'[ G is a 2..connected graph of diameter at most 3 that contains no induc::d subg~'aph isomorphic to K1.3 or B and G contains a vertex adjacent to exac:ly two w,rtices of degree 2, then G is hamiltonian. RemarL 3. The supposition that the diameter be 3 or less in Theorem 4 cannot be weake~e'l The family of graphs shown in Fig. 5 is constructed by taking a copy of K,~ (m >/3) and a copy of Kn (n/> 3) and joining vertices with the paths shown. These £r.~phs are not homogeneously traceable as x is not the initial vertex of a hamiltf.;aian path. However, these graphs have diameter 4 and C~ntain no induced K~,3 c:~rb . x
Fig. 5. We also note that 2-connectedness is necessary in Theorems i, 3 and 4. The graph F of Fig. 2(a) contains none of the forbidden subgraphs of Theorems 1, 3 and 4, but is not traceable.
Conclusion We feel that 'forbidden subgraphs' offer an interesting approach to 'hamiltonian' problems. We would like to point out some possible directions.
196
R.J. Gould, M.S. lacobson
The graph Kt,3 plays a m a j o r role in the results of this p a p e r as well as those in [3-7]. Can results be found that do not use K~.~? Perhaps Kt,. (n > 3) can be of some he,lp. In [711, Oberly and Sumner and [6] K a n e t k a r and Rao combine forbidden subgraphs with local connectivity. Can forbidden subgraphs be combined with other degree restrictions to yield new results? A simple alteration to Fig. l(b) shews that the d i a m e t e r restriction of T h e o r e m zi cannot be dropped. Can a reasonable set of forbidden subgraphs b e found that eliminates the need for this restriction?
Acknowledgement The authors would Hike to thank the referees for their suggestions, which considerably aided i,~ tile clarity of the exposition.
~'.efer,ences [1! M. Behzad. G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs (Prindle, Weber & Schmidl, Boston, 1979). [2] G. Chartrand, R.J. Gould and S.F. Kapoor, On homogeneously traceable nonhamiltonian graphs, Ann. New York Aead. Sci. 319 (1979) 130-135. [3] D. Duffus, R.J. Gould and M.S. Jacobson, Forbidden subgraphs and the hamiltonian theme, Proc. 4|h Int. Conf. on the "l'laeoryand Applications of Graphs, Kalamazoo, 1580 (Wiley, New York, 1981) 297-316. ['~.] S. Goodman and S. Hedetniemi, Sufficient conditions for a graph to be hamiltonian, J. Combin. Theory (B) 16 (1974) 175-180. [5] R.J. Gould, Traceability in graphs, Doctoral Thesis, Western Michigan University, 1979. [6] S.V, Kanetkar and P.R. Rao, Connected: locally 2-connected, K~.3-free graphs are panconnected, 2. Graph Theory, to appear. [7] D. Oberly and D. Sumner, Every connected locally connected nontrivial graph with no induced claw is hamiltonian, J. Graph Theory 3 (1979) 351-356.
:,
,:
: :
189
FORB~DEN S~GIIAPFtS AND EIAMILTONIAN PROPERTIES OF GRAPHS Ronaid J. G O U L D *
Emory University, Atlanta, GA 30322, USA Michael S. JACOBSON**
university of Louisville, Louisville, KY 40292, USA Received 24 November 1980 R¢vised 14 September 1981 and 21 December 1981 ~¢adous sufficientconditionsare giveii,in terms of forbiddensubgraphs, that implya graph is ei~hc~homogeneouslytraceable, hamiltoni~nor pancyclic. We consider only finite undirected graphs without loops or multiple edges. No~atie, or terms not defined here can be found in [1]. Let G be a graph and Iet S ~ V(G). The subgmph (S) induced by S is the graph with vertex set S and who~t~ edge set consists of those edges of G incident with two vertices of S. The distence d(u, v) between vertices u and v in a connected graph G is the minimum number of edges in a u - v path. The diameter of a graph G is max .... yea) d(u, v). A graph is hamiltonian (traceable) if it has a cycle (path) contaipSng all its vertices. A pancyclic graph of order p conlains a cycle of length l for eac~ ,~ (3 <~l ~
TheoremA
[4]. If G is a 2-connected graph that contains no induced sr~bgraph isomorphic to KL3 or Z~, then G is hamiltonian. We note that the proof of Theorem A actually shows that either G is a cycle or * Research supported by a grant from Emory University. ** Research supported by a grant from the Universityof Louisville.
0012-365X/82/0000-0000/$02.75 © 1982 North-Holland
199
R.J.
Gould, M.S.
Ja~bson
G is pancyclic. We now show that slightly more general conditions yield the same result.
Theorem 1. I[ ,G is a 2-connected gral:h that contains no induced subgraphs isomorphic to Kr 3 or Z~, then G is a cycle or pancyclic°
Proo|. The result is trivial if ]V(G)I<~4. So ,,mppose G satisfies the hypothesis, is not a cycle, and ilas order at least 5. Let C: vov~ • • • vk-lVo ( k >I 2) be an arbitrary cycle of length k. We show that if V ( G ) ¢ V ( C ) , then a (k + 1)-cycle can be found. Since G is not a cycle and contains no induozd KL3, then it must contain a 3-cycle. Since G is not connected there e:tists Xoe V ( G ) - V ( C ) such that Xo is adjacent to a vertex of C. Without loss of generality we may assume Xo is adjacent to 01 (or else relabel C), There is no induced K~.3 in G so the edge roY2 is in G or a (k + l)-cycle would be produced. Since G is 2-connected, there exists an x 0 - v2 path P, not conlaining vl. Consider the subpath P': XoXl • • • xlvi where vj is the first vertex of C on P. Case 1. Supp~se vi : ~2 (or by symmetry vj = Vo). The graph ({VO, 1)1, D2, XO,X1})~-Z2 unless one of the edges XlVo, x~vl or xtv2 is present. Consider x~v2 (and note a similar argument holds for xlvo). If vk_l ='- v2, then xoxlv2v~xo is a cycle of length 4. If vk-~ ~ v2, then ({v~, v2, Va, xz}) implies that v~ v.a ~ E ( G ) . If 1~_~ = 03, then vovlxox~v2vo is a 5-cycle. If Vk-~ ~ V3, we consider ({v~, vl, v3, xo}). If xov3 is an edge of G, then v o v 2 x l x o v 3 V 4 " " v k - l V o is a ( k + l ) - c y c l e . Hence we see that VoV~ is an edge of G. The graph ({ vo, v2, v3, x t xo})----Z~_unless one of the edges XoV3, x~ vo or J:~v3 is in G. But the inclusion of any of these edges produces a (k + 1)-cycle. If xl v~ ~ E ( G ) , we may consider the path P": x~x2" • • x~v2 and repeat the above argument. Eventually, either the previous possibility results or a (k + 1)-cycle is formed. Case 2. Suppose 3 ~ f ~< k - 1. Since G contains no induced Z2, it is easily seen that for some i (0<~ i ~< l), v~x~ and x~vj are edges of G. Observe that v~# vi_~ and vk I ¢ v~ or a (k + 1)-cycle can be found. Further, oi_iv~+t ~ E ( G ) since G contains no induced KI.3. Consider ({v~,, vl, v2, x~, vi}). It follows tha~ either ViVo~ E ( G ) in wh~.ch case I)oVk_ t . . . .
V~+~l)i_ l ..-
VlXiViV 0
is a (k + 1)-cycle or v i v 2 ~ E ( G ) , and VoVk_t
...
vi+ivi_
~ ...
v2vixiv~v
o
is a (k + ][)-cycle. Thus viv~ ~ E ( G ) and ({v~, xi, v i, v2, v3})----Z2 unless at least one of x~v2, x, vs, viv2, t~iv3 or v~v3 is an edge of G. The first three yield immediate (k + l)-cycles. If viv3a E ( G ) , then port,_
~ • , . VI+~I)i_ ~ ,..
is a (k + l)-cycle.
I)3I)iXiD~I)2V 0
191
Forbiddensubgraphsand harniltonianproperties
Finally, ff v l v 3 e E ( G ) , then ({Co,el, v2, vj, v~-I})---Z2 nless one of VoVj, v2vj, VoOi-l, v2vi-1 or v1%-1 is ~n G. But each edge produce~ a (k + 1)-cycle and the result follows. [] Remark lo We note that the hypothesis of Theorem I does not imply that the graph is panconnected. Consider K, (n t> 3) with vertices x , x 2 , . . . , x,. Let G be that graph obtained by subdividing the edge xtx2 and name the new vertex x. Thi~ graph does not contain an induced Kt, a or Z2 yet there is no x - x 1 path of length 2. Furth~'rmore, we cannot omit either of the induced subgtaphs from the hypothesis. Fig. l(a) shows a nonhamiltonian graph with no induced Z2. It is constructed by taking two copies" of C2~÷~ ( n > 1) and joining corresponding vertice~ in each copy by a path of length 2. Fig. l(b) shows a nonhamiltonian graph w~th no induced KL3. It i~ constructed by taking two copies of K2n.~ (n > 1) and ioinlng corresponding vertices in each copy by an edge and a path of length 2.
(a)
Contains no
Z2 .
(b)
Contains
no
Kl,, ~ .
F i g . 1.
The following was shown in [3]. Theorem B. If G is a 2-connected graph that contains no induced subgraph isomorphic to Kt.3 or F (see Fig~ 2(a)), then G is hamiltonian. We note that the condition~ of this theorem are not enough to imply that the graph be pancyclic; for example the graph of Fig. 2(b) is 2-connected, contains no induced subgraph isomorphic to Kt,3 or F and is not pancyclic. However it is easily shown that the hypothesis of Theorem 1 implies the hypothesis of Theorem B. The proof is routine and not included. Proposition 2. If G is 2-connected and contains no induced subgraph isomorphic to KI,a or Z2, then G contains no induced subgraphs isomo~hic to F.
R.I. Gould, M.S. lacobson
192
T]~.~ graph
F .
(b)
A no'e.pancycllc g r a p h .
Fig. 2.
The graph in F~g. l(b) shows that Z~ cannot be replaced by Z3 in Theorem 1.. In The~rem 3 we raodify the set of forbidden subgraphs to include Z3, but the c~nclusion i~ weaker than ~hat of Theorem i. Let B be that graph obtained by k~entifying a vertex in two distinct copies of K 3. "l'heo~rem 3. I[ G is a 2..connected graph that contains no induced subgraph isomorphic ~c. K;,:~, B, or Z~, then G is hamiltonian. Proof. Suppo,,;e G satisfies the hypothesis and is not hamiltonian. Choose a cycle C: t~,,t,~ " " ~k~:o (k > 2) of maximum length in G. Let xo~ V ( G ) - V ( C ) such ~hat x,, is adiaccnt w,, a vertex of C~ Without loss of generality suppose Xo is adjacent to tq. Siricc K~.3 is not an induced subgraph of G, V o v 2 e E ( G ) or a longer cycle would be present. Since G is 2-cop~nected, there must exist an x o - '02 path P that does r~ot contain t~. Suppose ~,/ is the first vertex of P on C. It is clear that t', , t~j. L~ E ( G ) and that j ~ 0, 2, 3, 4, k -- 1, or k, for otherwise C would not be of maximur,n length. Since G contains no induced Z3, we cant find a v ~ - v j path P', that is :~ ~ubpath of P. and is disjoint from V ( C ) - { v ~ , vj}. Further, the length of P' is a~ mc~st 3. Suppose the ~ength of P' is 3. that is, suppose P' is the path VtXoX~Vj. By considering ({v,~. v~, u:. x~, x~. t~i}),one can readily establish that v~v/is an edge of G. The subgraph ({t~, v2, J"o, t:/})~- Kt.3 unless v,_t) is an edge of G. But now the cycle l)~l~t~i Xi,Xll.~il)2L13
" " " tli
1U/~l/)kt~O
has length lenge~ than C which is a contradiction. Thcrefore we may assume that the length of P' is 2, that is, suppose P' is the path t, lx,¢o/. We tirst consider the graph ({vo, v~, v:,Xo, vt, v/.~}). We need only cow,sider whether ~ot), v~vj, vav i, t,,~e/, t, v~ vi~.~ or v2vi. ~ are edges of G. If any one of v,~v~, z~_v~, v~v~÷~, or v ~ v ~ is an edge of G, a cycle longer than C is immediately produced. If v~ v~ ~ E ( G ) , the~ ({v0, v~, vz, xo, v~})-~B, unless G contains an edge previously considered. Thus v~vj.~ is an edge of G and we next consider ({ro. t~, t,i,., t,j,z}), If v~v~.:~ is an edge of G, then either ({q-t, Xo, vi, Vi+2})~KI,3 or a Iongt~r cycle exists. The edge vov~ has already been handled; hence we conclude that t~ov/., is an edge of G. Now the graph ({v6, vl.2, vi+ ~, vi, v~-a})~-B
Forbidden subgraphsand hamiltonian properaes
193
unless an edge already analyzed exists in G. Thus, all cases produce a cycle longer than C, and hence we conclude that G is hamiltonian Q Remark 2. The hypothesis of Theorem 3 does not imply that of Theorem B (see Fig. 3). We also note that the hypothesis of "l~eorem 3 does not imply the graph is pancyclic (see Fig. 2(b)).
Fig. 3. A graph with no induced K1,3,Z3 or B that containsan induced F. Theorero C [5]. If G is a 2-connected graph o/diameter at most 2 that contains no indt:ce,~ :ubgraph isomorphic to Kl,3, then G is hamiltonian. We rJote that 2-connected nontraceable graphs of diameter 3 with no induced K:,:~ can be found (see Fig. 4). However, we may modify the set of forbidden subgra~hs to produce homogeneously traceable graphs of diameter at most 3.
K3n (n_>3)
Fig. 4. A graph with diameter 3 and no induced K1,3that is not traceable.
194
1~.3. Gould, M.S. 1acobson
Theorem 4. r[ G is a 2-cormected graph o f d i a m e t e r at m o s t 3 a n d G contains no i n d u c e d s.ub,graph isomorphic to Kt,3 or B, then G is h o m o g e n e o u s l y traceable. Proof. By Theorem C we need only cor~sider graphs of diameter 3. So suppose C satisfies the hypothesis of the Theorem but is not homogeneously traceable. Thus there exists a vertex vo that is not the initial vertex of a hamiltonian path in G. Let P: v o v ~ ' " v , be a longest path with initial vertex vo. Thus there exists x e V ( G ) - V ( P ) such that x is adjacent to a vertex v~ of P. Furthermore, we may as~urne i < n or a longer path would be evident and we may assume 0 < i or else v,, would be a cutvertex. Observe that v~..~v~+t ~ E ( G ) . Hence i < n - 1 . Since 2 ~- d(x, v,,) < 3 we consider two cases. C a s e 1. Suppose d ( x , vn) :: 2. Since v,, must be adjacent only to vertices of P, the vertex intermediate to x and v. on the distance path must lie on P, If XVoV. is the path of length 2, then ({x, Vo, vt, v . } ) ¢ K~,s implies that v~v. is an edge of O. It follows that the path voxv~v~+~ • • • v . v ~ v : • " • v~_t is longer than P and has initial ~:crtcx v~,. Thus we may assume xv~v. (0 < i < n) is the distance path, or we would merely change our choice of vi above. Now VoVj • • • v~-~vHvi+2 • " • l)ntliX is a path longer than P. Case 2. Suppose d(x, v . ) = 3. Let xv~qv, be the x - v . distance path. Clearly t~i. t~t ,5/9.
';ubcase 2a. Suppose j = 0. Since ({v0, v~, v~, 1)n})~K1,3 at least one of viva, v,v,, or v~v,, is an edge of G. Except for v~v. longer paths are easily established.
So suppose v~v. ~ E ( G ) and consider {{vo, v~, vi+~,x}). The edges xv~+~ and xvo yield longer paths easily; while if vovi+t ~ E ( G ) the path t'oU,, t l - ' i ~ 2 . . . . 1 . l n O l l ) 2 ' ' " 1)i_t/)iX is a longer path. 5 u b c a s e 2b. Suppose 0 < i < i - 1. By considering ({v i, vi+~, vi, v.}) and ({t'j,t~ j , v . v , , } ) we must have v~vj_~ and vi÷Lv, as edges of G. Since ({v,, t'~ ~. t',, q , ~ . v . } ) ~ B at least one of r~,i_~vj?~, vi_~v., v~vi+~ and v~v. is an edge of G. However. a longer path results in each case. S u b c a s e 2c. Soppose j = i - 1. The path vov~ " • • v~_tv.v._~ " • • vix is a longer path. S~bcase 2d. Suppose j : = i + l ~ n - - 1 (j=n-1 lis Subcase 2f). Since ({r,.vi,l,t,~e,v.})~K~,3 we can conclude that v.~'~i+z~E(G). But since ({v, t, v~, t , ~ , v~.., v , , } ) ~ B at least one of vi_~vi+~, vi-~v., viv~.,z or viv. is in G. Again a longer path is easily established in each case. S u b c a s e 2e. Suppose i + 1 < j < n - 1. The argument for this case is analogous to that of Subcase 2b. .~ubcase 2f. Suppose j = n - 1 . With this being the final possibility we may assume for each longest path P of length n with initial vertex Vo and final vertex t',, and arly vertex x not on P, that d(x, v . ) = 3 and x v ~ v . _ t v , is a dista.:ce path where 1 ~ i ~< n - 2. Since G is 2-connected v. is adjacent to some vertex of P, say Vk. Clearly k5e i - 1 or i. If 0 < k < i - 1 and since ({v~, v., v~+l, th:_l})~ K:l.3, then v,,t,~ t, v.vt,, ~ or v~_ ~,Vk+~ is in G. Longer paths are immediate for the first two; so
Forbidden subgraphsand hamiltonianproperties
195 ~4
suppose Vk+~Vk-I is an edge of G. Then 0.: VoVl " " • Vk- tOk+1 " " " VnVk is a longest path with initial vertex Vo. Further, x~ V(O) so the distance from x to on is 2. But this contradicts t h e fact that d(x, v,,) = 3. A similar argumen'~ applies when i < k < n - 1. Hence- we may additionally assume that the last vertex of every longest path with initial vertex vo is adjacent to Vo. Consider (~'.
I)OV 1 • . . V/_lVi+
l • . . ~)~_lV[X.
Now O is a longest path with initial vertex vo and end vertex x so that xvo~ E(G). The~ ({vo, x, vt, vn})~Kl.3 unless one oY xv~, xv. or v~v, is an edge of G. In any case a new path longer than O (or P) is apparent. [] Since aomogeneously traceable nonhamiltonian graphs have no vertices adjacent to t~/o or more vertices of degree 2, the following is immediate. C o m f i a ~ 5. ~'[ G is a 2..connected graph of diameter at most 3 that contains no induc::d subg~'aph isomorphic to K1.3 or B and G contains a vertex adjacent to exac:ly two w,rtices of degree 2, then G is hamiltonian. RemarL 3. The supposition that the diameter be 3 or less in Theorem 4 cannot be weake~e'l The family of graphs shown in Fig. 5 is constructed by taking a copy of K,~ (m >/3) and a copy of Kn (n/> 3) and joining vertices with the paths shown. These £r.~phs are not homogeneously traceable as x is not the initial vertex of a hamiltf.;aian path. However, these graphs have diameter 4 and C~ntain no induced K~,3 c:~rb . x
Fig. 5. We also note that 2-connectedness is necessary in Theorems i, 3 and 4. The graph F of Fig. 2(a) contains none of the forbidden subgraphs of Theorems 1, 3 and 4, but is not traceable.
Conclusion We feel that 'forbidden subgraphs' offer an interesting approach to 'hamiltonian' problems. We would like to point out some possible directions.
196
R.J. Gould, M.S. lacobson
The graph Kt,3 plays a m a j o r role in the results of this p a p e r as well as those in [3-7]. Can results be found that do not use K~.~? Perhaps Kt,. (n > 3) can be of some he,lp. In [711, Oberly and Sumner and [6] K a n e t k a r and Rao combine forbidden subgraphs with local connectivity. Can forbidden subgraphs be combined with other degree restrictions to yield new results? A simple alteration to Fig. l(b) shews that the d i a m e t e r restriction of T h e o r e m zi cannot be dropped. Can a reasonable set of forbidden subgraphs b e found that eliminates the need for this restriction?
Acknowledgement The authors would Hike to thank the referees for their suggestions, which considerably aided i,~ tile clarity of the exposition.
~'.efer,ences [1! M. Behzad. G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs (Prindle, Weber & Schmidl, Boston, 1979). [2] G. Chartrand, R.J. Gould and S.F. Kapoor, On homogeneously traceable nonhamiltonian graphs, Ann. New York Aead. Sci. 319 (1979) 130-135. [3] D. Duffus, R.J. Gould and M.S. Jacobson, Forbidden subgraphs and the hamiltonian theme, Proc. 4|h Int. Conf. on the "l'laeoryand Applications of Graphs, Kalamazoo, 1580 (Wiley, New York, 1981) 297-316. ['~.] S. Goodman and S. Hedetniemi, Sufficient conditions for a graph to be hamiltonian, J. Combin. Theory (B) 16 (1974) 175-180. [5] R.J. Gould, Traceability in graphs, Doctoral Thesis, Western Michigan University, 1979. [6] S.V, Kanetkar and P.R. Rao, Connected: locally 2-connected, K~.3-free graphs are panconnected, 2. Graph Theory, to appear. [7] D. Oberly and D. Sumner, Every connected locally connected nontrivial graph with no induced claw is hamiltonian, J. Graph Theory 3 (1979) 351-356.