CONDITIONS FOR THE EXISTENCE OF HAMILTONIAN CIRCUITS IN GRAPHS BASED ON VERTEX DEGREES

Transcription

1 CONDITIONS FOR THE EXISTENCE OF HAMILTONIAN CIRCUITS IN GRAPHS BASED ON VERTEX DEGREES AHMED AINOUCHE AND NICOS CHRISTOFIDES 1. Introduction The terminology used in this paper is that of [7]. The term graph denotes a finite, undirected graph without loops or multiple edges. For any vertex x e V of a graph G = (V, E) let N(x) be the set of vertices adjacent to x, and let d{x) be the degree of x. For any two non-adjacent vertices a and b we associate the vertex set T = V {a, b} N(a) U N(b), and if necessary to avoid confusion we shall write T ab instead of T. Also let X ab = \ N(a) n N(b). We denote a(g) as the independence number of G. Let d 1 ^ d 2 <... ^ d n be the degree sequence in V, and let df ^ d$ ^... ^ d^w\ be the degree sequence of vertices in a subset W <= V, where all degrees are with respect to the graph G. A subset W of V is defined to be a 'good' subset in G if d$ > i for all w t e W. Clearly W\ ^ n 2, and every subset of a 'good' subset is a 'good' subset. A non-hamiltonian graph G is maximal if the addition of any edge transforms the graph into a hamiltonian one. The following proposition will be useful. PROPOSITION. Let W be a 'good' subset of V and q ^ n. In any graph G = (V, E) the following conditions (a) and (b) are equivalent. There exists an ordering x x, x 2,..., x n of the vertices of G such that Proof. (a)=>(b) The vertices in W are labelled w x,w 2,...,w\ w \, so that ^df^... ^af wl ; set x t = w t (i = 1, 2,..., W\). Now clearly d{x t ) ^ i and j) <j imply that x {, x } $ W; therefore it follows from (a) that jc f x^e implies that (b)=>(a) Let JC 15..., x n be an ordering which satisfies (b). Let / be the smallest index for which d(x t ) ^ / and given this /, lety be the smallest index satisfying (b). Then, clearly W ={x h \h <j,h^ i) is a 'good' subset and from (b), we have that d( Xi )+d( Xj ) > q. 2. Main results THEOREM 1. Let G = (V, E) be a graph and let W be a 'good' subset of V. If d(a) + d(b) ^ nfor any two non-adjacent vertices a, b in V W, then G is hamiltonian. Received 20 May 1982; revised 12 December 1983 and 20 March Mathematics Subject Classification 05C45. This work was partly supported by the SERC. /. London Math. Soc. (2) 32 (1985) JLM 32

2 386 AHMED AINOUCHE AND NICOS CHRISTOFIDES Proof. Let H be a maximal non-hamiltonian graph satisfying the condition of Theorem 1. Clearly H is not complete. Let a, b be two non-adjacent vertices, which implies that they are joined by a hamiltonian a-b path fi = aa 2 a 3... a n _ x b in H. We prove that a, b can always be chosen as elements of V W. Suppose, on the contrary, that at least one end point of n, a say, is an element of W. Clearly any vertex of R(a) = {a t a i+1 e N(a)} is a possible end point of a hamiltonian path finishing at b. Thus R(a) s w. Among all these possible end points of n we may assume that d(a) = max xer(a) d(x). This is a contradiction since max^g fl(a) d(x) > R(a) = d(a) by the definition of W. Therefore a, b are elements of V W. In H we have that ab $ E implies that d(a) + d(b) < n. The theorem is then proved by contradiction of the hypothesis. Bermond [4] gave a result which is equivalent to Theorem 1, and can be obtained by using the above proposition. By appropriately choosing W, we can easily derive the well-known theorems of Bondy, Posa, Ore, Erdos-Gallai which can be found in [3]. Note that Chvatal's well-known theorem on hamiltonian graphs [8] neither dominates nor is dominated by Theorem 1. We denote by K v the complete graph on p vertices and by K p its complementary graph composed of/? isolated vertices. We write G x U G 2 for the union of two graphs G x and G 2, and G x + G 2 for the join of the two graphs [7]. In a graph G = (V, E), let H = aa 2 a 3... a n _ t b be a given a-b hamiltonian path, and let r, s be integers such that r = min(k\a k en(b)} and s = max{k\a k en{a)}. Let A = {aj\j < r}, B = {^\j > s} and L = N(a) n N(b). A set : {aj I a j+1 e N(a t ) if j > i, a t _ x e N(a t ) if j ^ 0 of vertices is associated with a vertex a { of //. Note that a t e /?(fl<). THEOREM 2. Le/ G = (V, E) be a maximal non-hamiltonian graph; let W be any 'good' subset of V, and suppose that xy$e If there exists at least one pair a, b of non-adjacent vertices such that d(d) + d(b) ^ n 1, then G is isomorphic to the graph (K P U K q U K t _^) + K t, where p, q, t are positive integers. Proof The vertices of W are labelled w lt w 2,..., W\ W \ so d? < d$ ^... < dj» w \. The non-adjacent vertices a, b are joined by a hamiltonian path // = a l a 2...a n with a = a x,b = a n, since G is maximal. Clearly the set R(b) n N(a) is empty, for otherwise G would contain a hamiltonian circuit. Furthermore, a$r(b) U N(a) and therefore R(b) and N(a) form a partition of V a. Similarly, R(a) and N(b) form a partition of V b. Hence, and the following holds for each vertex a i of fj.: n-\o\ + \T\ = X ab, (1) and a j+l en(a), j I (2) and o, R(a) => a } e N(b). J

3 EXISTENCE OF HAMILTONIAN CIRCUITS IN GRAPHS 387 Moreover, the vertices a r,a 8 are elements of L = N(a)()N(b), A-aczN(a) and B b c N(b) for otherwise (2) would be violated. The next important step consists in proving that T, L form a partition of V-(A U B) and each vertex a t of T implies a^el and a j+1 el. (3) In fact, it suffices to prove that ^ e T implies that a i+1 e L since then, by similarity, fljef implies that a^xel and, given these implications, the sets T,L obviously form a partition of V {A U B) since neither two vertices of L nor two vertices of T are consecutive vertices along //. Let S = {a i et\a i+l $L}. (Note that from (2), a k es implies that a k+1 en(a) and a k+2 en(a).) Suppose that S # 0, for otherwise (3) is proved. We may assume that r > 2, for if r = 2 then the circuit a 2 a 3... a k+1 aa k+2... ba 2 would be hamiltonian (a k es). Hence d(a) ^ 2 + X ab, since at least two vertices a 2, a k are elements of N(a) but not elements of L = N(a) 0 N(b). First, we prove that for all a t et, we have R(a t ) Tf\ W. Assume that the statement is false for some a^e T, and let a f er{aj) n {N(a) U N(b)}. We may assume, without loss of generality, that/>y. Either the circuit if a f N(b), or the circuit aa 2... ty a f+1 a f+2... a n _ x ba f a f _ x... a j+1 a aa 2... ^_ x ba n _ x... a f+1 a t a j+1 if a f en(a), is hamiltonian, and therefore,...a f a ^T foralla f er. (4) From (1) and (4) we get d{a t ) ^ 1 + T\ = X ab for all a t et. As d(a) + d(b) = n-1 and d(a) ^ Aab + 2, we derive d(a^) + d(b) <n 2 for all a t et. This result implies that T' <=,W. More precisely, we have T={w lt w 2,..., w m }, (5) for otherwise, if T contains some vertex w t ew with i>\t\ then max j/et^(.y) > ^1 = ^ab by definition of W. This contradiction proves (5). Secondly, let R = R(a 2 ), R x 0 A and R 2 = R-R v We prove that R 2 T-S and a 2 e W. Assume that this statement is false and let a p er 2 (p > r by definition of R 2 ), and a p $ T. As r > 2, a 3 must be adjacent to a, but then either the circuit if a p e N(b), or the circuit aa 2 a p+1... a n _ t ba p a p _ x...a z a aa p a p _ x...a r ba n _ x...a p+1 a 2 a z...a r _ x a if a p en(a), is hamiltonian. (Recall that A a <= N(a).) Hence R 2 T. Now suppose that there exists k > r such that a k er 2 C\ S. We already know that a k+1 en(a), a k+2 en(a) whenever a k es. Thus there exists an a k+1 -a k+2 -p&th containing the vertices of A since aa r _ x e E. By replacing a k+1 a k+2 with this path in the circuit a r a r+1...a n _ x ba r we get a hamiltonian circuit in G, a contradiction. Therefore R 2 o S = 0 and the following holds: R 2^T-S. (6) It is clear from (2), (6) and the definition of R, that R c R(a), which implies that N(a 2 ) c= N(a). Furthermore at least two vertices a k+1, a k+2 are adjacent to a but not 13-2

4 -388 AHMED AINOUCHE AND NICOS CHRISTOFIDES to a 2. Thus d(a 2 ) ^ d(a)-2 and d(a 2 ) + d(b) < n 2, which implies that a 2 ew. It follows from (5) that there exists an integer j>\t\ such that a 2 = w t. Every vertex of R x can be chosen to be the second vertex along a new hamiltonian a-b path. The new path would be aa g a g _ 1...a 2 a g+1...a r...a n 1 b if a Q er v Therefore we may assume that R^aWand a 2 = w p with j'^\t\ + \R l \. On the one hand, and by definition of W, we get d(a 2 ) > /?J + T\. On the other hand, d(a 2 ) = 1 +1 R x +1R 2 by definition of R x and R 2 ; hence d{a 2 ) <\ + \R x \ + \T\ from (6). This last contradiction proves (3). To complete the proof, note that no edge a u a v with a u ea,a v eb exists in G, for otherwise aa 2... a u a v a v+1... ba v _ x... a u+1 a would be a hamiltonian circuit. By setting A = p, B = q, L = / and by using the fact that N^) c L for all o c e T, which is implied by (3) and (4), it is clear that Gc(^u^U^ H)+^ Tnis g ra P h is obviously non-hamiltonian and the proof is complete. A graph G is called a 1-tough graph [9] if for every proper subset S of V we have w(g S) ^ S\, where w(g S) is the number of components left by the removal of the set S of vertices. COROLLARY 1. Let G be a \-tough graph and let W be any 'good' subset of V. If d(a) + d(b) ~^n I for any two non-adjacent vertices a, b in V W then G is hamiltonian. Proof. This is immediate. COROLLARY 2. graph. If Let x x, x 2,..., x n be any ordering of the vertices of a l-tough then G is hamiltonian. Proof. This is an equivalent form of Theorem 2 by the above proposition. COROLLARY 3. Let G be a maximal non-hamiltonian graph. If i <j, d t ^ /, d i Kj^di + dj ^ n-1, /Proof This is immediate. COROLLARY 4. If for a graph G, i <\{n \)=>d t > i then G is hamiltonian or G (2K p U K t _ x ) + K t, in which case n is odd. Proof. If n is even then Corollary 4 is a well-known result by Posa [12]. If n is odd, the conditions of the corollary are satisfied. The rest of the proof is simple and can be found in [1].

5 EXISTENCE OF HAMILTONIAN CIRCUITS IN GRAPHS 389 COROLLARY 5. Let G = (V, E) be a non-hamiltonian graph. If a\d) + d(b) ^ n 1 for any two non-adjacent vertices a, b, then either (i) Gg(^u KJ + K X (G is not 2-connected), or (ii) G c A^D+ i (n+1) (n is odd). Proof. Set W = 0 in Theorem 2. If T = 0, then we have (i). If T # 0, then it contains vertices with degree greater than %(n 1), in which case a(g) > ^(«+1). On the other hand, G clearly possesses a hamiltonian path and therefore <x(g) ^ («+ 1). Hence a(g) = \(n-\-1) and (ii) holds. It is interesting to consider those graphs satisfying Ore's condition (that is, did) + d(b) ^ n for all non-adjacent vertices a, b) in the light of Theorem 2. The following corollary generalises a result by Lesniak [11]. COROLLARY 6. Let G = (V, E) be a graph satisfying Ore's condition. Then one of the following holds. (For definitions see [3].) (i) G is \-edge hamiltonian, hamiltonian-connected, I-hamiltonian, (ii) (K p U (iii) Kfr^ Proof. The proof is an easy consequence of Theorem 2 and its corollaries. COROLLARY 7. A graph G is hamiltonian if d x > 2 and d 2 =... = d n ^ \(n 1). COROLLARY 8. An h-regular graph is l-edge hamiltonian ifn^ 2h. A generalisation of Theorem 2 concerning maximal circuits is given below. The proof is on similar lines to that of Theorem 2 and can be found in [1]. THEOREM 3. Let G = (V, E) be a l-tough graph and let W be a "good" 1 subset of V. Ifd(a) + d(b) ^ q 1 for any two non-adjacent vertices a, b in V W, where q ^ n, then G contains a circuit of length at least q. A generalisation of a result by Dirac [10] and the well-known theorem by Chvatal [8] is the following. THEOREM 4. Let G = (V, E) be a maximal non-hamiltonian graph. If dj ^j ^ Qp-l) => d n+1^ ^ n-j then G is isomorphic to the graph (K p U K q U K t _^) + K t, where p,q,t integers and q e {1, p). are positive Proof Let the vertices in G be labelled according to an ascending order of their degrees and consider the following cases. (i) dj >j for ally ^ \n 1. In this case, Corollary 4 applies and G is isomorphic to the graph (2K p U K t _ l )-\-K t with p ^ 1, / ^ 1. (ii) d s = s for some s < \n 1. Choose s to be the maximum integer such that d 8 = s. Let V x = {v t / ^ s}, V 2 = { Vi \i^n+l-s}. Clearly V 1 \ = \ V 2 \ = s. We first

6 390 AHMED AINOUCHE AND NICOS CHRISTOFIDES prove that for all v e V 2, d(v) = n \. Assume that the last statement is false and let / be the smallest value index such that v t e V x is not adjacent to some Vj e V 2. Using the conditions of Theorem 4 we have </(ty)+</(y fc ) ^ n for all k ^ s and thus i < s. The vertex v i must then be adjacent to all v k with k > i and thus d(vj) ^ n 1 i. It is clear that {u l5 v 2,..., yj is a 'good' subset since / < s and hence d(vi) > i. Therefore d{v^ + d(v } ) ^ n, which contradicts the assumption that ViV^E, and hence the statement d(v) = n \ for all ye V 2 is true. By setting t = s,p = \V\ \V 1 \ \V 2 \ we see that G is isomorphic to the graph (K p U K t ) + K t, and the proof is now complete. By setting W = 0 in Theorem 2, the following stronger result is obtained. (The proof is given in [1].) THEOREM 5. Let G (V, E) be a 2-connectedmaximal non-hamiltonian graph. If d(a) + d{b) ^ n 2for any two non-adjacent vertices then G is isomorphic to one of the following graphs: n odd, n even, = Aj (n _ a) + (% n _ 2) U K 2 ), n even, = K 2 + 3/LQ, n = Conjectures CONJECTURE 1. Let G be a 1-tough graph let W be any 'good' subset. If d(a) + d(b) + d(c) ^ n for any three mutually non-adjacent vertices a, b, c with a, be V W and c that vertex with the smallest degree (amongst the vertices non-adjacent to a or b), then G is hamiltonian. In [5] the following theorem of Jung is given. IfG is a \-tough graph of order n ^ 11 andd(a) + d(b) ^ n 4for any two non-adjacent vertices a, b then G is hamiltonian. This result would be a simple corollary of Conjecture 1 (in the particular case where W = 0). The conjecture given below generalises and strengthens Theorem 2. CONJECTURE 2. Let G be a l-tough graph and let W be any 'good' subset of V. If d(a) + d(b) ^ q 2 for any two non-adjacent vertices a, b in V W, where q ^n is an integer, then G contains a circuit of length at least q. If true, this conjecture will give the best possible result if q < n. This is clear from the graph G* defined as follows. Take 3 vertex-disjoint cliques of order r $s 2, add a vertex which is joined to every other vertex and three edges xy, yz, zx where x, y, z are three vertices, each one chosen from a different clique. The graph obtained is l-tough. It is easy to check that the maximal circuit has a length 2r+2 = 2d l -\-2.

7 EXISTENCE OF HAMILTONIAN CIRCUITS IN GRAPHS 391 References 1. A. AINOUCHE, 'Connectivity, independent sets and maximal circuits in undirected graphs', Ph.D. Thesis, London University A. AINOUCHE and N. CHRISTOFIDES, 'Strong sufficient conditions for the existence of Hamiltonian circuits in undirected graphs', J. Combin. Theory Ser. B 31 (1981) C. BERGE, Graphs and hypergraphs (North-Holland, Amsterdam 1973). 4. J. C. BERMOND, 'On Hamiltonian walks', Proceedings of the fifth British combinatorial conference (Bell, Winnipeg 1976) pp B. BOLLOBAS, Extremal graph theory (Academic Press, London 1978). 6. J. A. BONDY and V. CHVATAL, 'A method in graph theory', Discrete Math. 15 (1976) J. A. BONDY and U. S. R. MURTY, Graph theory with applications (MacMillan, London 1975). 8. V. CHVATAL, 'On Hamiltonian ideals', J. Combin. Theory Ser. B 12 (1972) V. CHVATAL, 'Tough graphs and Hamiltonian circuits', Discrete Math. 5 (1973) G. A. DIRAC, 'Note on Hamilton circuits and Hamilton paths', Math. Ann. 206 (1973) L. LESNIAK, 'n-distant Hamiltonian graphs', Utilitas Math. 9 (1976) L. PosA, 'A theorem concerning Hamilton lines', Magyar Tud. Adak. Mat. Kutako Int. Kozl 7 (1962) Department of Management Science Imperial College London SW7 2BX