Kotzig, Anton

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Matematický časopisAnton KotzigOn Decomposition of a Tree into the Minimal Number of PathsMatematický časopis, Vol. 17 (1967), No. 1, 76--78Persistent URL: http://dml.cz/dmlcz/126915Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1967Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czON DECOMPOSITION OF A TREE INTO THE MINIMAL NUMBER OF PATHS ANTON KOTZIG, Bratislava Throughout the paper we mean by a graph a non-oriented finite graph. The object of our investigation will be trees i. e. connected graphs not con-taining any circle ( = Kreis [1]). Our considerations are necessarily based on some known lemmas, that is why we shall have to mention them (without proofs). Lemma 1. ([1], p . 49.) Any tree ivith at least one edge contains at least two vertices the first degree. Lemma 2. ([1], p . 21.) The number of vertices of odd degree is in each graph even. Lemma 3. ([1], p . 22.) Let G be any connected graph and let 2n (where n > 0) be the number of its vertices of odd degree, then there exists a decomposition of the graph G into n open moves (move = Kantenzug [1]) and any decomposition of the graph G into open moves contains at least n moves. Apart from the above, we have: Lemma 4. Let G be any tree, then there does not exist in G any closed move with at least one edge and every open move in G is a path (== Weg [1]). Proof . The validity of the lemma is evident from the fact tha t any tree does not contain a circle. Lemma 5. Let G be any tree and let 2n (n > 0) be the number of its vertices of odd degree, then G may be decomposed into n paths and any decomposition of the graph G into paths contains at least n paths. Proof . From Lemma 1 it follows tha t n ^ 1. Hence it follows from Lemma 3 tha t G may be decomposed into n open moves and from Lemma 4 it follows that each such open move is a path. From the above the validity of the first assertion of the lemma is evident. From Lemma 3 and from the fact tha t n > 0 it follows tha t each decomposition of the graph G into paths contains at least n paths. The proof lemma is accomplished. Let us now put the following question: How many different decompositions 76 of the given tree into the minimal number of paths do there exist? The fol-lowing theorem solves the problem: Theorem. Let G be any tree with at least one edge and let d(i) be the number of its vertices of i-th degree. Let 2n be the number of the vertices from G that are of odd degree (i. e. 2n = d(\) + d(3) + .. .) and let r be the number of different decompositions of G into n paths, then we have r = f[gW», i=l where for every natural i we put g(2i —1) = g(2i) = 1 . 3 . 5 (2i —1) . Proof . Let V = {v ±, v2, • • •, vm} be the set of vertices of G. By Hi we denote the set of all edges from G incident at v% (i = 1 , 2 , . . . , m). Let Ri be any de-composition of the set Hi with following property: if \H%\ = 0 (mod 2), then each class from Ri has exactly two elements and if \H%\ = 1 (mod 2), then one of the classes of Ri contains an only edge (it will be called the significant edge with respect to Ri; if Hi contains an odd number of edges then none of its edges is significant to Ri) and the other classes of the decomposition con-tain two elements each. With regard to the system R = {i?i, R2, ..., Rm} of decompositions with the above property the following evidently holds: each vertex and only vertex of odd degree vj is incident at such an edge and only at one such an edge that is significant with respect to Rj e R. Let us travel along the elements of G according to the following rules: (1) If in a travel we arrive along an edge / at is end ( = vertex vx), then we proceed along that edge which with / forms a 2-element class of Rx e R. If, however, the edge / is significant with respect to Rx, we finish our travel-ling in vx. (2) We start each of our travels in a vertex vu of odd degree along such an edge from Hu that is significant with respect to Ru e R. I t is evident that the elements covered at any of these travels form a path of G, whereby the starting (as well as the final) vertex is a vertex of the odd degree. Any edge from G belongs evidently to one of the n paths describing all such travels (if, of course, we do not take into consideration in which of the two possible directions we travel). Hence: To each system R = {R±, R2, ..., Rm} of decompositions with the required property there corresponds (uniquely) a decomposition of the graph G into n paths. To the different systems there correspond different decompo-sitions of G into n paths. I t follows that r is equal to the number of different systems R with the required property. Then the validity of the theorem be-77 comes evident from the fact that g(s) is the number of the different decompo­sitions of the set of all edges incident at the given vertex of 5-th degree with the required property as well as from the fact that the decomposition of such a set may be chosen for the individual vertices quite independently. R E F E R E N C E [1] K ó n i g D., Theorie der endlichen und unendlichen Graphen, Leipzig 1936. Received April 2, 1966. Katedra mimerickej matematiky a matematickej statistiky Přírodovědecké] fakulty Univerzity Komenského, Bratislava 78 

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On Decomposition of a Tree into the Minimal Number of Paths

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On Decomposition of a Tree into the Minimal Number of Paths

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