# Graphs with 1-Factors

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2 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 1, January 1974 GRAPHS WITH 1-FACTORS DAVID P. SUMNER ABSTRACT. In this paper it is shown that if G is a connected graph of order 2n (n > 1) not containing a 1-factor, then for each k, 1 <k_n, there exists an induced; connected subgraph of order 2k which also fails to possess a 1-factor. Several other sufficient conditions for a graph to contain a 1-factor are presented. In particular, it is seen that the connected even order line graphs and total graphs always contain a 1-factor. In this paper all graphs will be finite, undirected and without loops or multiple edges. A 1-factor of a graph is a spanning subgraph which is regular of degree one. Tutte [5] has characterized graphs which contain 1-factors. Beineke and Plummer [1] proved that every block with a 1- factor always contains at least one more, and a result due to Petersen [4] showed that every cubic graph with no bridges contains a 1-factor. Our purpose in this paper is to show (Theorem 2) that for any connected graph of order 2n which does not contain a 1-factor and for each integer k, 1 <k<n, there is an induced, connected subgraph of order 2k which also fails to possess a 1-factor. Moreover, several other sufficient conditions for a graph to contain a 1-factor are given. In particular, we see that the connected, even order line graphs and total graphs must always contain a 1-factor. Our terminology will conform to that in Harary [3]. DEFINITION. If x is an endpoint of the graph G and x is adjacent to z, then we will say that z is the joint of x. If x and y are two endpoints of G which have the same joint, then we will call x and y coincident endpoints. We note that no graph with coincident endpoints can possess a 1-factor. LEMMA 1. If G is a connected, nontrivial graph with no coincident endpoints, then there exist adjacent points x, y E G such that G-{x, y} is connected. PROOF. Since the lemma clearly holds for complete graphs, we assume that G has diameter d>2. Let a, y E G be points of G which are a distance d apart and let D=a a... xy be a path of length djoining a andy. Suppose Received by the editors September 5, AMS (MOS) subject classiications (1970). Primary 05C99. American Mathematical Society 1974

3 GRAPHS WITH 1-FACTORS 9 that G* =G- {x, y} is not a connected graph. Let A be the component of G* that contains a. We note that since D is a diameter of G, every point in G* -A is adjacento x. Thus if G* -A contains a nontrivial component B, then for any b, c E B which are adjacent, every point in G -{b, c} is joined to x by a path, and hence G-{b, c} is connected. Similarly, if there exists a point e E G*-A which is adjacent to y, then G-{e, y} is connected. Hence we may suppose that G* -A contains only isolated points and each of these points is adjacent only to x. Thus every point in G*-A is an endpoint of G with joint x. Therefore since G has no coincident endpoints, G* -A must consist of a single isolated pointf If y were an endpoint of G, thenf and y would be coincident endpoints. Hence we may assume that y is adjacent to some element of A, and so G-{f, x} is connected. C1 THEOREM 1. If G is a connected graph of even order, then G2 has a 1-factor. PROOF. Noting that the result holds for graphs of low order, we proceed by induction. Suppose that G is connected of even order and that the theorem holds for graphs of smaller order. By Lemma 1 we may find points x, y E G such that d(x,y)_2 and G-{x,y} is connected (for we may either choose x and y to be coincident endpoints or, if none such exist, we may. take x and y to be the points guaranteed by Lemma 1). Thus (G- {x, y})2 has a 1-factor, which together with the edge xy of G2 yields a 1-factor for G2. 0 Thus since the total graph T(G) of a graph G satisfies T(G)= [S(G)]2 (Behzad[2]) where S(G) is the subdivision graph of G, we have COROLLARY 1. Every connected, total graph of even order has a 1-factor. THEOREM 2. If G is a connected graph of order 2n (n >1) and k is an integer, 1 <k_n, such that every induced, connected subgraph of order 2k has a 1-factor, then G has a 1-factor. PROOF. We proceed by induction. The theorem is easily checked for n=2, 3. Suppose that it holds for some n-i (n_4). Let G be a connected graph of order 2n and suppose k is such that 1 <k_?n and every induced, connected subgraph of order 2k contains a 1-factor. Clearly we may assume that k< n- 1 the case k=n being trivial. If G contained two coincident endpoints, then we could extend the graph consisting of these two points and their common joint to a connected, induced subgraph of order 2k which has no 1-factor. Hence G does not contain two coincident endpoints and thus by Lemma 1, there exist adjacent points x, y E G such that G-{x, y} is connected. Thus by our inductive assumption, G-{x, y} has a 1-factor which together with the edge xy constitutes a 1-factor for G. El

4 10 D. P. SUMNER [January COROLLARY 2. If G is a connected graph of even order with no induced K1,3, then G has a 1-factor. PROOF. The only connected graph of order four which does not have a 1-factor is K1 3. Thus (except in the trivial case where G is K2) the corollary is the special case of the theorem with k=2. D Since no line graph can contain K1,3 as an induced subgraph, we obtain COROLLARY 3. Every connected line graph of even order has a 1-factor. A 1-factor of a graph may be viewed as a partition of the points into two-element subsets in such a way that the elements of each set are adjacent. As a result of Corollary 3, we see that it is always possible to form a similar partition of the edges as long as there are an even number of them. Similarly, as a result of Corollary 1, if a graph contains an even number of elements (i.e., points and lines), then we may partition the elements into subsets of order two so that the elements of each subset are either adjacent or incident. As another immediate consequence of Corollary 2 we have COROLLARY 4. Every connected cubic graph in which every point lies in a triangle has a 1-factor. REMARK. An alternate proof of Corollary 4 is obtained by noting that in any such graph other than K4 the collection of those edges that lie in an even number of triangles constitutes a 1-factor. As an example, consider the graph in Figure 1. The edges which are numbered determine such a 1-factor ~~~47 FIGURE 1 Our next result is an extension of Theorem 2. By an end-block we mean one which contains exactly one cutpoint. THEOREM 3. Let G be a connected graph of even order. If (i) G has exactly one block of even order, and (ii) for each block B of G, there exists an integer k, 3 <k< jbj such that every induced, connected subgraph of B having order k contains a 1-factor, then G has a 1-factor.

5 1974] GRAPHS WITH 1-FACTORS 11 PROOF. If G has exactly one block, then G has a 1-factor by Theorem 2. Suppose G is a graph satisfying the conditions (i) and (ii) such that the theorem holds for graphs with fewer blocks. Since G has at least two endblocks, there exists an end-block B of odd order. Let b E B be the unique cutpoint of G lying in B, and let k be the integer associated with B. Then B-{b} is connected of even order and every induced, connected subgraph of B-{b} having order k has a 1-factor (note k< IBI-1 since IBI is odd). Thus B-{b} has a 1-factor by Theorem 1. Also (G-B)U{b} has fewer blocks than G and satisfies (i) and (ii) and hence has a 1-factor. Thus we obtain a 1-factor for G. D The next theorem may be proven in a similar manner. THEOREM 4. If G is a connected graph of even order such that (i) G has exactly one block of even order, and (ii) no block of G contains an induced K1,3, then G has a 1-factor. THEOREM 5. If G is a connected graph of even order and every induced K1,3 contains a bridge whose deletion results in two components of even order, then G has a 1-factor. PROOF. We induct on the number of points in G noting that the result holds for small orders. Let G be a graph satisfying the hypotheses of the theorem and such that the theorem holds for graphs with fewer points than G. If G has no induced K1 3, then G has a 1-factor by Corollary 2. If G does not contain an induced K1,3, let e=ab be an edge of this K1,3 such that the deletion of e results in a graph having the two even components A and B with a E A and b E B. Suppose A contains an induced K1,3. Letf be an edge of this subgraph such that G-{f} contains two components of even order. Since both endpoints off lie in A, A-{f} consists of two components C and D. We may assume a E D. Then C is one of the components of G-{f} and hence has even order (as must D also). Thus A satisfies the conditions of the theorem and so must have a 1-factor. Similarly, B contains a 1-factor which together with that for A produces a 1-factor for G. E Since this paper was submitted, we have learned that our Theorem 1, Corollary 1, and Corollary 3 were discovered independently by G. Chartrand, A. Polimeni, and J. Stewart and occur in their paper The existence of a 1-factor in line graphs, squares, and total graphs, which is to appear in Indagationes Mathematicae. ACKNOWLEDGMENT. Thanks are due to Frank R. Bernhart for pointing out to me an error in a previous version of this paper.

6 12 D. P. SUMNER REFERENCES 1. L. W. Beineke and M. D. Plummer, On the 1-factors of a nonseparable graph, J. Combinatorial Theory 2 (1967), MR 35 # M. Behzad, A criterion for the planarity of the total graph of a graph, Proc. Cambridge Philos. Soc. 63 (1967), MR 35 # F. Harary, Graph theory, Addison-Wesley, Reading, Mass., MR 41 # J. Petersen, Die Theorie der reguldren Graphen, Acta Math. (1891), ,W. T. Tutte, The factorizations of linear graphs, J. London Math. Soc. 22 (1947), MR 9, 297. DEPARTMENT OF MATHEMATICS, UNIVERSrrY OF SOUTH CAROLINA, COLUMBIA, SOUTH CAROLINA 29208

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