Cycles of given length in some K1,3-free graphs

Discrete Mathematics 78 (1989) 307-313 North-Holland

307

CYCLES OF GIVEN LENGTH IN SOME &,-FREE GRAPHS Gun-Quan ZHANG Department of Mathemutim, West Virginia University, Morgan&own WV 26506, U.S.A. Received 2 March 1987 Revised 31 July 1987 Let G be a non-trivial connected &,-free graph. If any vertex cut of G contains a veitex v such that G@!(u)) is connected, we prove that G is pancyclic. If G(Z+I(u))is conaected for any vertex u of G, we prove that G is vertex pancyclic and obtain a polynomial time algorithm for constructing cycles of given length and passing through a given vertex in G.

1.IntrodractIon A graph G is called Ki,&?ee if no induced subgraph of G is isomorphic to K1,3. A vertex v of G is called locul~y connected if the induced subgraph G@!(V)) is connected. A graph G is called local/y connected if every vertex of G is locally connected. A graph G is called quarilocully connected if every vertex cut of G contains a locally connected vertex. Obviously, a connected locally connected graph must be quasilocally connected. But the converse it not true and an example can be found in Fig. 1. A graph G is said to be pancyclic if G contains cycles of all possible lengths. And G is said to be vertex pancyclic if each vertex of G is contained in cycles of all possible lengths. In (21, Oberly and Sumner proved that a connected, locally connected K&_ree graph contains a Hamiltonian cycle. Clark [2] improved Oberly-Sumner’s result, and discovered that a connected, locally connected K1,3-free graph is pancyclic. In this paper, we will prove that a quasilocally connected K,,+ee graph is pancyclic and a connected, locally connected K&ree graph is vertex pancyclic. Actually, a stronger result will be given for the quasilocally connected Ki,+ee graphs, which is similar to the property of vertex pancyclic. In [I, 31, Minty and Sbihi obtained a polynomial time algorithm to find the maximum independence number of a K &ree graph. In this paper, we will give a polynomial time algorithm to construct a cycle of given length containing a given vertex in a connected, locally connected K,,,-free graph. The distance between vertices x and y in a subgraph H is denoted by d&, y). The diameter of a subgraph H is the maximum distance between any pair of vertices in H. If C is a cycle of a connected graph G, then any component B of 0012-365X/89/$3.50 @ 1989, Elsevier science Publishers B.V. (North-Holland)

C.-Q. Zhang

308

Fig. 1.

G\V(C) is called a bridge of C. And N(B) n V(C) is called the attachment of the bridge B.

2. Main results Lema 1. Let G be a K1,3-free graph. If v isa locally connected vertex of G, then the diameter of the induced subgraph G(N(v)) is a most 3.

Proof. Assume that there are two vertices x and y in N(v) such that the shortest path P = uIu2. . . ut joining x = u1 and y = u, in G(N(v)) is of length at least four, i e. t 35. Since G is K&ree and G(v, ul, u3, u,) cannot be a K1,3 subgraph, there must be an edge joining two vertices of {u,, u3, u,} which contradicts the path P being the shortest. Cl eorem 2. Let G be a K1,3-fiee graph and C be a cycle of length r in G where r c IV(G)l. If C has a bridge B such that the attachment of B contains a locally connected vertex, then V(C) is contained in a cycle of length r + 1. roof. LetC=v,...

v,vl and v1 be a locally connected vertex contained in the attachment of a bridge B. Let W = V(G)\V(C). Assume that V(C) is not contained in any cycle of length r + 1. I. For any ViE V(C), we ciaim that if z E N(r),) n W for any ViE V(C), then C”i-l,

vi+l

) E E(G),

(vi-19

Z)

Q E(G)

and

(Vi+19

Z)

$ E(G).

If z E N(Vi) 17-W, then (vi-l, Z) and (vi+15z) $ E(C?

because otherwise

Cycles in some K, ,3-free

309

V i_lZV~.. . Vi+1 or ViZVi+l.. . Vi would

be a cycle of length r + 1 containing V(C). Since G is K1,3-free and G(vi, Z, Vi+l, Vi-l) cannot be an induced & graph, we must have (Vi-19Vi+l)E(G)* II. Let M = G(N(v,)). By Z and Lemma 1, 2~ d&, N(v,) (I W and k E (2, t}.

z) s 3, where z E

HZ. If z E N(v,) n W and d,(v,, z) = 2 (k E (2, r}) then V(C) is contained in a cycle of length r + 1. Indeed, without loss of generality, assume k = 2 (the case k = r is symmetric). Let y E N(v,) be such that zyv2 is a path in H. If y E W, then v1yv2.. . vl would by a cycle of length r + 1 containing V(C). So assume that there exists i E {3,4, . . . , r - 1} such that y = vi. Since (z, Vi) E E(G) we have, by I, that (vi-l, vi+l) E E(G) and ~2 . . . vi,lvi+l. . . v,vlZviv2 would be a cycle of length r + 1 containing V(C). IV. Let R=v2... z be the shortest path in H joining v2 and z for some z E N(v,) n W. We have, by II and 111, that IV(R)1 =4 and V(R)\{v,, Z) c V(C). Let R = V2ViVjZe

We claim that i #j f 1. Suppose i = j + 1. Then v2 . . . VjZVlVi. . . v,v2 would be a cycle of length r + 1 containing V(C) which contradicts the assumption. (The case i=j - 1 is symmetric.) V. Since z E N(tJj) n W, we have, by I, that (vi-19 Vj+l) E E(G). We wish to show that (Vi-l, vi+l) $ R(G). Suppose (Vi-l, Vi+l) E E(Gj. Then the cycle

i>j, or ~2...Vi-1Vi+l... v,vlZvjviv2when j > i would be of length r + 1 and contains V(Cj.

212.. . Vj-lVj+l

l

Vj-lVj+l

l

l

l

l

l

Vi-lVi+l

l

l

l

Vu,VlZVjViV2 when

Since G is &-free and {Vi, vl, Vi-19 Vi+l} cannot induce a & graph, either (vl, vi+l) E E(G). We shall consider the following four cases.

(v,, vi-l)or

Case 1. (v,, Vi-l)E E(G) and i > j + 1. The cycle ~2. . . vj-lVj+l. . . Vi,1v1ZvjVi.. . V,V~ would be of length r + 1 and contains V(C). Case 2. (~1, Vi+l)E E(G) and i j + 1. The cycle ~2. . . vj-lvj+l . . . ViVjZVlvi+l. . . V,V~ would be of length r + 1 and contains V(C). Case 4. (v,, Vi-l)E E(G) and i < j - 1. It is similar to Case 3.

C.-Q. Zhang

310

The proof of the theorem is complete.

Cl

Theorem 3. If G is a connected, locally connected &-free graph of order n, n > 2, then each vertex of G is contained in a cycle of length h, h = 3, . . . , n. proof, First notice that the minimum degree of G is at least 2. Indeed, if (x, y) E E(G) and d(x) = 1, then d(y) = 1 because otherwise G@!(y)) would not be connected. This contradicts that G is connected and n > 2. Hence, IN(v)1 2 2 and G(N(v)) being connected implies that v is contained in a cycle of length 3 for any v E V(G). By Theorem 1, v is contained in cycles of all possible lengths. Cl Thmmm 4. Let G be a quasilocally connected K&?ee graph of order n, n > 2. If the vertex v of G is contained in a cycle of length r, then v is contained in a cycle of length h, for any h = r, r + 1, . . . , n. Proof. We will prove this theorem by induction on the number of vertices. When G = K,, the theorem is obvious. Assume that the theorem is true for n - 1 (n 3 4). Let G be a quasilocally connected K1.&ee graph of order n. Assume that G is not a complete graph. vu,vl be a cycle of length r containing the vertex Letr
l

v1

IZL We shall show that IN(u) n V(C)1 s 1 for any u E W. for i = 1, . . . , r (mod r), we can have the two paths P =v2u4v6. . . Q. . . v2r and Q = ulv3us. . . Q+~. . . ‘u~,,-~ where y is the greatest integer such that y s (r/2). The segment of P between Vi and vi is denoted by UiPUj if i c j or UiPUjif i > j. If IN(u) n V(C)1 2 2, let ul, uh E N(U). Then the cycle C’ = Since (Vi--1, Vi+,)

E

E(G)

Cycles in some K,,3-j?ee

311

V~UV~CV,V2PV . v h_2@l when h is odd, or C’ = v1uv,,Cv,v2Pv~-2v,,-,~v, Id_1 when h is even would be of length r + 1 which contradicts the assumption.

IV. Let ViE V(C). Since {Vi, Vi+19 z, z ‘} does not induce a K1,3 graph for any z, z’ E N(vi) fTIV, (z, z’) E E(G). Hence, N(vi) KW induce a clique in G for any Vi E V(C)* V. We claim that v1 is not a cut-vertex of G(N(x)) for any locally connected vertex x of G. Suppose x. E A@,) and x is a locally connected vertex of G, then x E W by I. If G(N(x)\{v,)) 1~ lr) disconnected, let N(x)\(v,} = X U Y where X and Y are two disconnected parts of N(x). X U Y c W because IN(x) n V(C)1 s 1 (by III) and vl E N(x) n N(C). Since G(N(k)) is connected and v1 is a cut-vertex of G(N(x)), there exist vertices zl and z2 such that z1E X, z2 E Y and (v,, zl), (vl, z2) E E(G). By IV, N(v,) n W is a clique and therefore (zl, z2) E E(G), which contradicts that X and Y are disconnected and our claim follows. VI. We claim that G ’ = G\{v,} is quasilocally connected &-free graph. It is obvious that G’ is &-free. Let U be a vertex cut of G’. Then either U or U U {v,} is a vertex cut of G. Since vl is not locally connected, U contains a locally connected vertex x in G. By V, x remains as a locally connected vertex in G’ because v1 cannot be a cut-vertex of G(N(x)) and G’(N(x)\{v,}) is still connected. VII. If n - lar+ I. Since v,v2.. . l-rr is a cycle of length r - 1 in G’, by induction, G’ has a cycle of length P + 1 containing v, which is also a cycle in G. Hence, nsr+l, i.e. n = r + 1 and W is a single vertex. This leads to a contradiction that the attachment of W is V(C) and IN(W) n V(C)1 = 1 (by III). So we have proved the theorem. Cl Tkorem 5 If G is a qzuzsilocallyconnected &&ee then G contains cycles of all possible lengths.

graph of order n, n > 2,

Proof. Assume that G is not a complete graph. Let X, y be a pair of non-adjacent vertices of G. Since V(G)\{x, y} is a vertex cut, there always exist a locally connected vertex v in G. Then there is a cycle of length 3 in G({v} U N(v)). By Theorem 4, the vertex v is contained in cycles of all lengths. 0 An example which is quasilocally connected but not locally connected can be found in Fig. 1. (The vertex v in Fig. 1 is not locally connected.) Theorem 6. If G is a connected, locally connected &-free graph and C = VI. . . vu,vl ii a cycle of length r in G, then the following algorithm can be,!:ti-=+

C.-Q. Zhang

312

construct a cycle of length r + 1 which contains V(C), where r C !V(G)l. This algorithm is of polynomial time.

Algorithm. Step 1. C +{q,

W t V(G)\V(C). Step 2. Choose vi E C such that N(q) n W # 0. Relabel {v,, . . . , v,} such that v, +v~+~__~,for t = 1,2, . . . , r (mod r). A dV(v,) f3 C, B +N(vl) n W, A+ 4V(v2) f3A. If (u,, v,) $ E(G), then go to 3, otherwise go to 4. Step 3. Choose z 5 N(Q) n W. If (v,, z) E E(G), then C’ +v,zv, . . . v,, otherwise C’+v1zv2. . . v,q. Go to 8. Step 4. If B n N(A+) = 8, then go to 5. Let z E B n N(A+) and let vi E A+ n N(Z). If (z+_~,vi+*) $ E(G), then relabel {v,, . . . , II,} such that r (mod r), and go to 3. ur + Vi+r+rfort=l,..., C’ +V2 a Vi-IVi-1 V,V1ZVJjtJ2. Go to 8. Step 5. Let vi E N(A+) n N(Z) n A. If (vi_19 v~+~)$ E(G), then relabel {v,, . . . , v,} such that 21,+Vj+,-l for t = 1, : . . , r (mod r), and go to 3. Step 6. Let ViE N(vj) n A+. Ifi>j, thensti, tcjandgoto7. If i < j, then relabel {v,, . . . , v,) such that vh +vr+&_h for h = 1, . . . , r (mod r), ands+r+2-i, t+r+2-j (modr), and go to 7. SQ 7. If (vS+ q+L) E Z(G), then C’ +x5. . . v,_gJ,+1 . . . Vs-_lVs+l . . . ~u,ug.J,vsv2 and go to 8. If (v,+ v,) E E(G), then C’ +2/z. . . v,_1zI,+1 . . . vs-gJ1zz/,‘u,. . . ?J,‘u2 and go to 8. . . . , q},

l

C’

+?J2

l

. . . u,_gJ,+1

l

l

l

. . . UJ&3J12’,+1

. . . ?J,v2.

Go to 8. Step 8. C’ is the cycle length r + 1 and contains C. END. of. See the proof of Theorem 2.

Cl

Cycles in some K,,3-jkee

313

Fig. 2.

Finally, the author would like to ask the following question: Conjecture. If G is a 3-connected Kl,S-free graph such that each vertex cut is not an independent set, then G contains a Hamilton cycle. The conjecture would not be true, if the graph is only 2-connectea (see Fig. 2).

References G.J. Minty, On maximal independent sets CFvertices in claw-free graphs. J. Combin. Theory, Ser. B, 28 (1980) No. 3, pp. 284-304. D. Oberly and D. Sumner, Every connected, locally connccteC nontrivial graph with no induced claw is hamiltonian, J. Graph Theory, Vol. 3, (1979) pp. 351-356. N. Sbihi, Algorithme de recherche d’un stable de cardinalit maximum dans un graphe sans Ctoile, Disc. Math. 29 (1980) No. 1, pp. 53-76.