Let G=(V, E) be a 2-connected graph. We call two vertices u and v of G a K4-pair if u and v are the vertices of degree two of an induced subgraph of G which is isomorphic to K4 minus an edge. Let x and y be the common neighbors of a K4-pair u, v in an induced K4−e. We prove the following result: If N(x)N(y)N(u)N(v){u,v}, then G is hamiltonian if and only if G+uv is h amiltonian. As a consequence, a claw-free graph G is hamiltonian if and only if G+uv is hamiltonian, where u,v is a K4-pair. Based on these results we define socalled K4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K4-closures

Broersma, H.J.

English

Broersma, Haitze J.

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A note on K4-closures in Hamiltonian graph theory

University of Twente Research Information

Discrete Mathematics 121 (1993) 19-23 North-Holland I9 A note on &-closures in hamiltonian graph theory H. J. Broersma Department of’ Applied Nether1and.s Received I5 November Revised 24 April 1991 Abstract 1990 Broersma, H.J., A note on K,-closures in hamiltonian graph theory, Discrete Mathematics 121 (1993) 19-23. Let G =( V, E) be a 2-connected graph. We call two vertices u and c of G a K?-pair if u and L’ are the vertices of degree two of an induced subgraph of G which is isomorphic to K, minus an edge. Let x and y be the common neighbors of a K,-pair u, c in an induced K,-e. We prove the following result: If N(x)uN(y)~N(u)uN(v)uju, c}, then G is hamiltonian if and only if Gfuc is hamiltonian. As a consequence, a claw-free graph G is hamiltonian if and only if G + UC is hamiltonian, where u, L’ is a K,-pair. Based on these results we define socalled K,-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K,-closures. 1. Introduction We use Bondy and Murty [4] for terminology and notation not defined here and consider simple graphs only. About 15 years ago the now well-known paper ‘A method in graph theory’ by Bondy and Chvhtal [3] was published. The closure concept they introduced in this paper opened a new horizon for the research on hamiltonian and related properties. We focus on the hamiltonian problem and mention their main results. In the sequel let G be a graph on n vertices, and let d(x) denote the degree of the vertex x of G. Lemma 1 [3]. Let u and G’ be distinct nonadjacent vertices in G such that d(u) + d(v) 3 n. Then G is hamiltonian if and only if G + uv is hamiltonian. Correspondence to: H.J. Broersma, Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands. 0012-365X/93/$06.00 C) 1993- -Elsevier Science Publishers B.V. All rights reserved Bondy and Chvatal defined the n-closure C,(G) of G as the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n until no such pair remains. They showed that C,,(G) is well defined and that G is hamiltonian if and only if C,(G) is hamiltonian. As a consequence, if G is a graph on at least 3 vertices and C,,(G) is complete, then G is hamiltonian. Inspired by these results several other closure concepts were developed (e.g. in [l, 2, $6, 8-121). Most of these concepts are based on conditions for nonadjacent pairs of vertices which include some ‘global’ parameters of the graph, i.e., the number of vertices or the cardinality of a maximum independent set of vertices containing the pair. Recently, Hasratian and Khachatrian [7] introduced a closure concept based on ‘local’ conditions concerning the extended neighborhood structure. Here we discuss a condition which takes into account the local structure of the graph. 2. Main results Denote by K,-e the complete graph on four vertices minus an arbitrary edge. A pair of vertices (u, c) s V(G) is a K,-pair of G if u and L’ are the two nonadjacent vertices in an induced subgraph H of G which is isomorphic to K,---e; the pair of vertices of degree 3 in H is called a copuir efu and L’. By N(u) we denote the set of neighbors of a vertex L’ of G. Theorem 2. Let (u, U) he a K4-pair of G with copair (x,1,) such that Nun N(u)uN(u)u(u, c.). Then G is hamiltonian !f and only [f G+uc is hamiltonian. Proof. If G is hamiltonian, then obviously G+ UL: is also hamiltonian. Conversely, assume G + uu is hamiltonian and G is not hamiltonian for some K,-pair {u, U) with copair (x, yi satisfying N(x)uN(y) c N(u)uN(u)u iu, ui. Then G contains a Hamilton path ~:ic~ . . . c,_ 1c, with ~‘i = u and o, = L‘. Without loss of generality assume x = Ui and JJ = L’~ for some i and .i with 2 < i <j < n - 1. Clearly i <j- 1, otherwise cicz “‘CiL‘,u,_i “‘I’if1Ui is a Hamilton cycle in G. By similar arguments ui+ ,#N(u) and Z’j- i&V(u). Hence i <,j- 2, 2’i + 1 EN(C) and ~!j_ IEN( But now U1Uj_lUj~2"'Gi+,C,C‘,~1 "'ajUiUi_, "'U1 is a Hamilton cycle in G, a contradiction. z Theorem 2 has the following consequence for clu\v$ree graphs, i.e., graphs that do not contain an induced subgraph isomorphic to K,,,. Corollary 3. Let [u, c) he a K,-pair qf a clawfree graph G. Then G is hamiltonian (fund only if G + UC is hamiltonian. 21 Fig. 1. G, Proof. Let {u, 21) be a K,-pair of a claw-free graph G with copair {x, y}. If won (or N(y)), then w~N(u)uN(u)u{u,v), otherwise the subgraph of G induced by (w,.x, U, v> (or {w, y, U, u}) is isomorphic to K1,3. Hence, N(x)uN(y)cN(u)uN(v)uCu,o), and the result follows from Theorem 2. 0 Note that, in fact, we only need that the subgraphs of G induced by (N(x)u(xj)- {y) and (N(y)u{y))-(x) are claw-free. Corollary 4. Let {u, v) be a K4-p&r qf G such that (N(u)uN(o)l =n-2. Then G is hamiltonian $’ and only if G + uu is hamiltonian. Proof. Let (u,v) be a K,-pair of G with IN(u)uN(u)( =n -2. Then N(u)uN(r)= V(G)-{u,u}, and obviously any copair jx,y] of {~,t;) satisfies N(x)uN(y)~N(u) uN(o)u(u,~‘). The result follows from Theorem 2. q Based on the above results, we say that a graph H is a K,-closure of G if H can be obtained from G by recursively joining K,-pairs that satisfy the condition of Theorem 2 and H contains no such pairs. A graph can have different K,-closures. Consider, e.g., the graph G1 of Fig. 1. It is easy to check that we can recursively join the K,-pairs {2,6), { 1,3), {l, 5), {l, 43, {2,5), {3,6), {4,7}, {2,4} and {4,6) to obtain K7 as a K,-closure of G,, whereas, on the other hand, we can recursively join the K,-pairs (1,3}, { 1,5), {1,4], {2,5), {3,6) and {4,7) to obtain K, minus the edges ofa triangle as another K,-closure of G1. Although a unique K,-closure may not exist, obtaining a K,-closure of G can be helpful to answer the question whether G is hamiltonian. The following results are obvious consequences of Theorem 2. Theorem 5. Let H be a K4-closure qf‘ G. Then G is hamiltoniun if and only if’ H is hamiltonian. Corollary 6. Let G be a graph on n33 vertices. If K, is a K,-closure of G, then G is hamiltonian. 3. Applications In this section we consider some classes of graphs that have K, as a K,-closure. 22 k Zk, ka2 T,, k>l Fig. 2. ak+z L a+~, k>O Proposition 7. Let G be a connected graph with complete subgraphs H 1, H,, , H, such that: (i) E(G)=E(H,)uE(H,)u ... uE(H,), (ii) V(Hi)nV(Hj)=~ or jV(HJnV(HJ32 (1 <i<j,<t), (iii) V(Hj)nV(Hj)nV(H,)=@ (1 di<J<k<t). Then K, is a K4-closure qf G. Proof. By induction on t. For t = 1 the statement is trivial. Assume the statement is true for t<s and let G be a graph satisfying the conditions of the proposition with t =s+ 1 > 2. Since G is connected, there exist i and .i such that Hi and Hj have a nonempty intersection. By (ii) and (iii), V(Hi)nV(Hj)nV(G) contains a pair (x,yJ such that N(X)UN(y) C V(Hi)U V(Hj) C N(U)UN( ) v U{U, U} for all (u,u} with uEV(Hi) and ~czP’(H~). Using Theorem 2 we can recursively join all nonadjacent pairs {u,I;) with copair {x, y) to obtain a complete subgraph Hij of the new graph G’ with V(Hij)= V(Hi)U V(Hj). It is not difficult to see that G’ is a connected graph with complete subgraphs Hi, Hi, . . , Hi satisfying (i), (ii) and (iii). By the induction hypoth- esis, K, is a K,-closure of G’ and hence of G. 0 Examples. Fig. 2 shows some examples of classes of graphs that can be shown to be hamiltonian using Corollary 6 and Proposition 7 (or an algorithm based on these results). Note that one third of the vertices of the graph Zk, k 3 2 has degree 2, that the diameter of the graph L2*, r > 3 is r and that the maximum degree of this graph is 5. If wedeletetheedgesoftheedgeset [ii,i+l)li=1,3,...,2k+l) fromLZk+6,k>0,then the remaining graph has maximum degree 4 and also has KZk+6 as a K4-closure. It is clear from the above examples that Corollary 6 can be useful in cases where counterparts of Corollary 6 based on degree conditions or neighborhood conditions are not useful. Therefore, it seems to be worthwhile to try to obtain more results on Hamiltonian graph theor? 23 local structure conditions. However, it is also clear that the above methods are not useful in general. Of course it is possible to combine several closure conditions to obtain more powerful concepts, but which combination gives ‘the best’ concept? Is there a unifying condition that covers all known closure conditions? References [I] A. Ainouche and N. Christofides, Strong sufficient conditions for the existence of hamiltonian circuits in undirected graphs, J. Combin. Theory Ser. B 31 (1981) 339-343. [Z] A. Ainouche and N. Christofides, Semi-independence number of a graph and the existence of hamiltonian circuits, Discrete Appl. Math. 17 (1987) 213-221. [3] J.A. Bondy and V. Chvital, A method in graph theory, Discrete Math. 15 (1976) I I l-135. [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976). [S] R.J. Faudree, R.J. Gould, M.S. Jacobson and L. Lesniak, Neighborhood closures for graphs, Colloq. Math. Sot. Jinos Bolyai 52 (1987) 227-237. [6] AS. Hasratian and N.K. Khachatrian, Some localization theorems on hamiltonian circuits, J. Combin. Theory Ser. B 49 (1990) 287-294. [7] A.S. Hasratian and N.K. Khachatrian, On stable properties of graphs, Discrete Math., to appear. [8] I. Schiermeyer, A strong closure concept for hamiltonian graphs, preprint. [9] 1. Schiermeyer, A polynomial algorithm for hamiltonian graphs based on transformations, Ars Combin. 25B (I 988) 55-74. [IO] Y.J. Zhu, and F. Tian, A generalization of the Bondy-Chvital theorem on the k-closure, J. Combin. Theory Ser. B 35 (1983) 2477255. [I 11 Y.J. Zhu, F. Tian and X.T. Deng, More powerful closure operations on graphs, Discrete Math., to appear. 1121 Y.J. Zhu, F. Tian and X.T. Deng, Further considerations on the BondyyChvital closure theorems, preprint. 

A generalization of the Bondy-Chvital theorem on the k-closure,

A method in graph theory,

A polynomial algorithm for hamiltonian graphs based on transformations,

A strong closure concept for hamiltonian graphs,

Graph Theory with Applications (Macmillan, London and Elsevier,

Neighborhood closures for graphs,

On stable properties of graphs,

Semi-independence number of a graph and the existence of hamiltonian circuits,

Some localization theorems on hamiltonian circuits,

Strong sufficient conditions for the existence of hamiltonian circuits in undirected graphs,

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A note on K4-closures in Hamiltonian graph theory

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