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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 721051, 6 pages
http://dx.doi.org/10.1155/2013/721051
Research Article

Decomposition of Graphs into Paths and Cycles

1Core Group Research Facility (CGRF), National Center for Advanced Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, Anand Nagar, Krishnankoil 626190, India
2Department of Mathematics, The Madura College, Madurai 625 011, India
3Department of Mathematics, Christ University, Bangalore 560 029, India

Received 31 October 2012; Revised 12 March 2013; Accepted 12 March 2013

Academic Editor: Kinkar Ch Das

Copyright © 2013 S. Arumugam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A decomposition of a graph is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . If each is a path or a cycle in , then is called a path decomposition of . If each is a path in , then is called an acyclic path decomposition of . The minimum cardinality of a path decomposition (acyclic path decomposition) of is called the path decomposition number (acyclic path decomposition number) of and is denoted by () (()). In this paper we initiate a study of the parameter and determine the value of for some standard graphs. Further, we obtain some bounds for and characterize graphs attaining the bounds. We also prove that the difference between the parameters and can be made arbitrarily large.

1. Introduction

Graph decomposition problems rank among the most prominent areas of research in graph theory and combinatorics and further it has numerous applications in various fields such as networking, block designs, and bioinformatics.

A decomposition of a graph is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . Various types of path decompositions and corresponding parameters have been studied by several authors by imposing conditions on the paths in the decomposition. It is obvious that every graph admits a decomposition in which each subgraph is either a path or a cycle. In this connection, Erds asked what is the minimum number of paths into which every connected graph on vertices can be decomposed and Gallai conjectured that this number is at most as stated below.

Gallai's Conjecture (see  [1]).  If is a connected graph on vertices, then can be decomposed into paths.

A good number of research articles have been published in which Gallai's is the focus of study and still this conjecture remains unsettled for more than 30 years. Towards a proof of the conjecture, Lovasz [1] made the first significant contribution by proving the following theorem.

Theorem 1 (see [1]). A graph on vertices (not necessarily connected) can be decomposed into paths and cycles.

Gallai's conjecture and Theorem 1 motivate the following definition.

Definition 2. Let be a decomposition of a graph . If each is either a path or a cycle, then is called a path decomposition of . If each is a path, then is called an acyclic path decomposition of . The minimum cardinality of a path decomposition of is called the path decomposition number. Similarly the acyclic path decomposition number of is defined and is denoted by .

The parameter was introduced by Harary and further studied extensively by Harary and Schwenk [2], Péroche [3], Stanton et al. [4, 5], and Arumugam and Suseela [6]. This paper initiates a study of the parameter path decomposition number.

2. Basic Terminologies and Results

By a graph we mean a finite, connected, and undirected graph without loops or multiple edges. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. The order and size of a graph are denoted by and , respectively.

If is a path in a graph , then the vertices are called internal vertices of and and are called external vertices of . For a cycle , the vertex is called the initial vertex and all other vertices are internal vertices. Two paths and of a graph are said to be internally disjoint if no vertex of is an internal vertex of both and . If and are two paths in , then the walk obtained by concatenating and at is denoted by , that is, . Further when , the path is denoted by .

A caterpillar is a tree in which removal of all the pendant vertices results in a path. For vertices and in a connected graph , the detour distance is the length of a longest path in . The detour diameter of is defined to be . Two graphs are said to be homeomorphic if both can be obtained from the same graph by a sequence of subdivisions of edges. The length of a longest cycle in a graph is called the circumference of .

We need the following theorems.

Theorem 3 (see [4]). For any tree , , where is the number of vertices of odd degree.

Theorem 4 (see [6]). Let be a unicyclic graph with cycle . Let denote the number of vertices of degree greater than two on . Let denote the number of vertices of odd degree. Then

Theorem 5 (see [2]). For the complete graph , one has .

Theorem 6 (see [2]). For the complete bipartite graph with , one has where is the conventional Kronecker delta.

3. Bounds for

In this section we obtain some bounds for the path decomposition number and characterize the graphs attaining some of the bounds obtained. Let us first establish an upper and lower bound involving the order of the graph and the number of vertices of odd degree. For this we define the term of a graph . Given a path decomposition for a graph , let , where denotes the number of internal vertices of . Now, define , where the maximum is taken over all path decompositions of .

Theorem 7. Let be a graph on vertices. Then , where is the number of vertices of odd degree in .

Proof. Let be the collection of all path decompositions of and let .
Then Hence, so that . Also, it is not difficult to see that and hence , which implies that . The upper bound follows immediately from Theorem 1.

Remark 8. If is a graph with vertices of odd degree, then if and only if there exists a path decomposition of such that every vertex is an internal vertex of members in .

The following examples show that there are infinite families of graphs attaining the bounds given in the above theorem and consequently prove that these bounds are sharp. That is, the family of all trees and an infinite class of unicyclic graphs achieve the lower bound, whereas some common classes of graphs such as complete graphs, wheels, and complete bipartite graphs of odd size attain the upper bound. In this connection we need to note down the inequality that which follows from the fact that every acyclic path decomposition of a graph is also a path decomposition.

Example 9. (i) For any tree , .
(ii) If is a unicyclic graph with cycle and if denotes the number of vertices on with degree greater than two, then

Proof. (i) Since every path decomposition of is an acyclic path decomposition of , we have and hence the result follows from Theorem 3.
(ii) If , then so that . If , then it follows from Theorem 4 that and hence . Further, for any path decomposition of all the odd vertices and at least one vertex on are external vertices of some paths in so that . Thus . If , by Theorem 4, we have and hence . Thus as .

Example 10. (i) For the complete graph , we have .
(ii) For the complete bipartite graph with
(iii) If denotes the wheel on vertices, then .

Proof. (i) Let be any path decomposition of . Since every member of covers at most edges, we have so that . Further, it follows from Theorem 1 that . Thus .
(ii) Assume first that . Now, if is odd, then every vertex is of odd degree and hence . If is even, then can be decomposed into hamiltonian cycles and hence . Further, for any path decomposition of , every member of covers at most edges so that and hence . Thus .
Suppose . If is odd, then every vertex is of odd degree and hence . If is even, then . Further, for any path decomposition of , every member in covers at most edges and hence . Thus .
(iii) Since the number of vertices of odd degree in is or accordingly as is odd or even, it follows from Theorem 7 that . Also by Theorem 1, we have and hence .

We now proceed to obtain some lower and upper bounds for involving detour diameter, circumference, and maximum degree of a graph along with the characterization of graphs attaining the bounds.

Theorem 11. For any graph , , where is the detour diameter of . Further, equality holds if and only if is a caterpillar with .

Proof. Let be a path in of length . Then is a path decomposition of and . Hence, .
Suppose . Let be a path of length in . If there is a vertex not on with , let be a longest path which contains and edge-disjoint from . Thus, , where is the set of edges not covered by the paths and is a path decomposition of and , which is a contradiction. Thus every vertex not on is a pendent vertex and hence is a caterpillar.
Now, if there exists a vertex on with , let and be two vertices not on which are adjacent to . Then , where is the set of edges not covered by the paths and is a path decomposition of and , which is a contradiction. Hence, , for all .
The converse is obvious.

Theorem 12. For any graph , , where is the circumference of . Further, equality holds if and only if is either a cycle or a cycle with exactly one chord or a unicyclic graph with and every vertex not on the cycle is a pendant vertex.

Proof. Let be a cycle of length . Then is a path decomposition of and . Hence .
Suppose . Let be a cycle of length in . We consider the following two cases.
Case  1  . We claim that has at most one chord. Suppose has two chords say and . If and , then , where is the set of edges not covered by and the path is a path decomposition of and , which is a contradiction. Suppose and . Let and denote the two -sections of the cycle and without loss of generality we assume that contains one of the -sections of . Now, let denote the path consisting of the edge , followed by the path and the edge . Then , where denotes the set of edges not covered by the paths and is a path decomposition of and , which is a contradiction. Hence, has at most one chord. Thus is a cycle or a cycle with exactly one chord.
Case  2  . Suppose that there exists a vertex not on with . Let be a longest path which contains and edge-disjoint from . Thus , where is the set of edges not covered by and is a path decomposition of and , which is a contradiction. Thus every vertex not on is a pendant vertex.
If there exists a vertex on with , let be a longest path containing and edge-disjoint from . Then , where is the set of edges not covered by and is a path decomposition of with , which is a contradiction. Hence every vertex on has degree less than or equal to .
Now, we claim that the cycle has no chord. Suppose is a chord of . Let be a pendant vertex not on . Then there exists a vertex on different from the vertices and such that and are adjacent. Let and denote the two -sections of and without loss of generality we assume that lies on . Let be the path consisting of the edge followed by the path and the edge . Now, , where is the set of edges not covered by the paths and is a path decomposition of and , which is a contradiction. Hence, has no chords.
Thus is a unicyclic graph with and every vertex not on the cycle is a pendant vertex. The converse is obvious.

Remark 13. Obviously, for any vertex of a graph , at most two of the edges incident with can be covered by a member of a path decomposition of . As this is true in particular for a vertex of maximum degree it follows that . This bound is sharp as a complete graph of odd order attains this. In fact, every graph which is hamiltonian cycle decomposable attains the bound. Further, for any tree , if and only if is homeomorphic to the star .

The following theorem characterizes the family of unicyclic graphs achieving the bound given in Remark 13.

Theorem 14. Let be a unicyclic graph with cycle and . Then if and only if every vertex not on has degree 1 or 2, at most two vertices on have degree and has exactly one vertex with degree if .

Proof. Let be a unicyclic graph with . Let be a minimum path decomposition of . Let be a vertex in with . Then every member of passes through .
We claim that lies on the cycle . This is obvious if . If , let and be the paths in covering the edges of the cycle . Since both and pass through ; it follows that lies on . Thus every vertex of degree lies on and consequently every vertex not on has degree 1 or 2.
Further if , it follows that contains exactly one vertex of degree and if , then contains at most two vertices of degree .
The converse is obvious.

In the following theorem we characterize graphs of order with and having minimum (maximum) number of edges.

Theorem 15. Let be a graph of order with . Then . Further, if and only if   is homeomorphic to the star . Also if and only if is hamiltonian cycle decomposable or is the complete graph with even.

Proof. Let be a graph with . Then is connected and hence .
Now, suppose . Then is a tree. As it follows that is homeomorphic to the star . Conversely, if is homeomorphic to , then .
Now, suppose . Then is -regular. If is even, then for any minimum path decomposition of , we have and each path in must cover edges so that is hamiltonian cycle decomposable. Conversely, if is hamiltonian cycle decomposable, then . If is odd, then and hence . Hence, and is even. The converse follows from Example 10(i).

4. Relation between the Parameters and

In this section we establish some relation between the parameters and .

Theorem 16. For any graph , one has . Further, if and only if every component of is a cycle.

Proof. The first inequality is already seen. For the other inequality, consider a path decomposition of and spilt each cycle member of into two paths and thus an acyclic path decomposition of with cardinality at most is obtained, which proves the required inequality. Now suppose . Let be a minimum path decomposition of . Then every member of is a cycle. Further, all the cycles in are vertex-disjoint and hence every component of is a cycle. The converse is obvious.

Remark 17. The first part of the inequalities given in the above theorem is strict in the sense that they both are equal for infinitely many graphs. For example, one has the following.(i)For even integers , we have , which follows from Theorem 5 and Example 10(i). (ii)Example 9 and Theorem 4 together show that the parameters and are equal for all trees and also for the class of all unicyclic graphs other than the cycles.

Though the upper bound suggested by Gallai's conjecture for and the upper bound given by Theorem 1 for differ by at most 1, the difference between the actual values of these parameters can be made arbitrarily large, which we proceed to prove. The following lemma is useful in this regard.

Lemma 18. Let be a connected graph which can be decomposed into a cycle and a path . Then can be decomposed into two paths if and only if has a vertex of degree 2 which lies on or on the -section of where is the smallest positive integer such that and and is the largest positive integer such that and .

Proof. Suppose can be decomposed into two paths and . Since and are the only two vertices of odd degree in , we may assume that is the origin of and is the origin of . Let be the terminus of . Since the degree of is even, it follows that is not an internal vertex of so that is the terminus of as well, and the degree of is 2. Clearly cannot lie on the -section or the -section of .
Conversely, suppose has a vertex of degree two satisfying the conditions stated in the lemma. We consider two cases.
Case  1  . Since is connected, . If , then trivially can be decomposed into two paths. Hence, we assume that . If there exists a -section of which is internally disjoint from , let be the path consisting of the -section of followed by the -section of which is not internally disjoint from and let be the path consisting of the -section of which is internally disjoint from followed by the -section of . Then is a decomposition of . The proof is similar if there exists a -section of which is internally disjoint from .
Now, suppose both the -sections of and both the -sections of are not internally disjoint from . Let be the largest positive integer such that and . Then the -section of and one of the -sections of , say , are internally disjoint. If has length greater than 1, then any internal vertex of has degree 2 and the -section of is internally disjoint from and hence the proof is complete. If the -section of has length greater than 1, let be the path consisting of the -section of followed by the -section of which is different from followed by the edge and let be the path consisting of the -section of followed by followed by -section of . Then is a decomposition of .
Case  2 (no vertex on has degree 2). In this case lies on the -section of . Let and let and be integers with such that and no internal vertex of the -section of lies on . Without loss of generality we assume that there exists a -section of , say , which contains and and let be the other -section of . Let be the path consisting of the -section of , followed by and the -section of . Let be the path consisting of the -section of followed by and the -section of . Then is a decomposition of .

Corollary 19. If is a graph admitting a minimum path decomposition having exactly one cycle , then . Further, if and only if there exists a path in with and the induced subgraph satisfies the conditions stated in Lemma 18.

Theorem 20. Given any positive integer , there exists a graph such that .

Proof. Let be the graph consisting of vertex-disjoint cycles and a - path , where , for . Clearly . Further, . We now prove that by induction on .
If , then it follows from Lemma 18 that . Assume that , for all graphs as described above with cycles. Let be such a graph with cycles. Let be an acyclic path decomposition of . Let be the path in containing the edge . Let , , and . Let be the paths in which decompose the graph and let be the paths in which decompose the graph . Clearly . We now claim that . Let be the end vertex of which lies in . Then the -section of together with the paths decompose the graph . Hence, by induction hypothesis so that . Hence, so that . Thus and so .

Theorem 21. Let be a connected graph. Then if and only if is the graph consisting of vertex-disjoint cycles and a path satisfying the following conditions. (i)Every vertex of the cycle , , lies on the path . (ii)If and denote, respectively, the smallest and the largest positive integer such that , then every edge of the -section of is a chord of the cycle .

Proof. Suppose . Let be a minimum path decomposition of . Then consists of exactly one path, say and vertex-disjoint cycles, say . Let . Since is connected, for all . Now, if there exists a vertex of degree 2 in which lies either on any of the cycles or on the -section of then it follows from Lemma 18 that , which is a contradiction. Hence, (i) and (ii) hold.
Now, the graph given in the theorem is nothing but the graph described in Corollary 19 with and thus the converse follows.

Theorem 22. Given two positive integers and with , there exists a graph such that and .

Proof. If , let be the graph with components, each component being a cycle. If , let be the graph given in Theorem 21. If , let be the graph consisting of exactly one cut vertex and blocks, each block being a cycle. In all these cases, and .
Now, suppose . Let , where . Let be the connected graph consisting of vertex-disjoint cycles and a path satisfying the following conditions: (i) if and is a proper nonempty subset of if ; (ii)if and denote, respectively, the smallest and largest positive integer such that , , then each edge of the -section of is a chord of the cycle .
Clearly and it follows from Lemma 18 that .

We conclude the paper by posing the following problem for further researchs. (i)Characterize graphs for which , where is the number of vertices of odd degree in . (ii)Characterize graphs for which . (iii)Characterize graphs for which . (iv)Characterize graphs for which .

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