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 1985. 1980 Mathematics Subject Classification 05C45. This work was partly supported by the SERC. /. London Math. Soc. (2) 32 (1985) 385-391 13 JLM 32
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
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 + 1 ^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
-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].
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
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 = 8. 3. 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.
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