On degree sum conditions for directed path-factors with a specified number of paths
Introduction
In this paper, we consider finite graphs. For terminology and notation not defined in this paper, we refer the readers to [3]. Unless stated otherwise, a graph (resp., a digraph) means an undirected graph without loops or multiple edges (resp., a directed graph without loops or multiple arcs). For a (di)graph , denotes the vertex set of , and is referred to as the order of . For a graph (resp., a digraph ), (resp., ) denotes the edge set of (resp., the arc set of ). A (directed) cycle-factor of a (di)graph is a spanning sub(di)graph of consisting of a union of vertex-disjoint (directed) cycles in . Note that a (directed) path-factor may contain (directed) paths of order one in the disjoint union of (directed) paths.
The Hamilton cycle problem, i.e., the problem of deciding whether a given graph has a Hamilton cycle or not, is a fundamental problem in graph theory. But, it is NP-complete, and so various kinds of sufficient conditions for hamiltonicity of graphs have been extensively studied (see survey papers [8], [9]). In particular, since the publication of the following theorem, various studies have considered degree conditions. Here, for a vertex of a graph , denotes the degree of in .
Theorem A Ore [14] Let be a graph of order . If for every two distinct vertices and with , then has a Hamilton cycle.
In 1997, Brandt et al. showed that the same degree condition as in Theorem A guarantees the existence of a cycle-factor with a specified number of cycles.
Theorem B Brandt et al. [4] Let be an integer with , and let be a graph of order . If for every two distinct vertices and with , then has a cycle-factor with exactly cycles.
Theorem A, Theorem B have been extended to the class of digraphs (of sufficiently large order) by Woodall (1972) and, Chiba and Yamashita (2018), respectively. Here, for a vertex of a digraph , and denote the out-degree and the in-degree of in , respectively.
Theorem C Woodall [17] Let be a digraph of order . If for every two distinct vertices and with , then has a directed Hamilton cycle.
Theorem D Chiba, Yamashita [7] Let be an integer with , and let be a digraph of order . If for every two distinct vertices and with , then has a directed cycle-factor with exactly directed cycles of order at least .
In 2013, Zhang et al. [18] characterized non-hamiltonian digraphs of order which satisfy for every two distinct vertices and with .
On the other hand, the Hamilton path problem is also NP-complete, and so we are also interested in degree conditions for the existence of a path-factor with a specified number of paths which is a general concept of a Hamilton path. The following is the path version of Theorem A, Theorem B.
Proposition E Let be an integer with , and let be a graph of order . If for every two distinct vertices and with , then has a path-factor with exactly paths.
This is a corollary of Theorem A (cf. e.g., Section 5.1 of [6]), but the degree condition is sharp. In 2005, Li and Steiner [13] characterized graphs of order which satisfy for every two distinct vertices and with and have no path-factors with exactly paths.
What happens if we do not allow paths of order one in a path-factor? In 2018, Chiba and Yamashita proved the following for connected graphs in a direct way by using a standard technique to show hamiltonian properties of graphs. (They actually considered the maximum value of the degree sums of two vertices in any independent set of size , see [6] for the details.)
Theorem F [6, Proposition 5.1.4] Let be an integer with , and let be a connected graph of order . If for every two distinct vertices and with , then has a path-factor with exactly paths of order at least .
In this paper, we consider paths of order at least for a fixed integer and extend Theorem F to the class of digraphs. The following is our main theorem. Here, a digraph is connected if the underlying graph is connected.
Theorem 1 Let and be integers with and , and let be a connected digraph of order . If for every two distinct vertices and with , then has a directed path-factor with exactly directed paths of order at least .
The degree condition is sharp. In fact, for the complete bipartite graph , does not have a directed path-factor with exactly directed paths, where denotes the digraph obtained from by replacing each edge in with two arcs and . On the other hand, the order condition comes from our proof techniques. Clearly, is necessary for the existence of the directed path-factor in Theorem 1. When is even, for the graph , the digraph satisfies the degree condition in Theorem 1, but it does not satisfy the conclusion in Theorem 1. Thus is necessary for the case where is even. Hence the order condition in Theorem 1 is best possible for .
This study is also concerned with the problem of finding a path-factor including all edges of a perfect matching in a bipartite graph. For a given bipartite graph with bipartition and a perfect matching of , if we replace each edge (, ) with an arc and contract all edges in , we get a digraph of order . In particular, an -alternating path (i.e., it is a path starting and ending with edges in such that the edges belong to and not to , alternately) of order in corresponds to a directed path of order in , and vice versa. This implies that Theorem 1 is equivalent to the following theorem (see also [7], [18] and Section 6.2 of [6] for the details of the equivalence and other related works). Here, for a bipartite graph with a perfect matching , an -path-factor of is a path-factor of consisting of -alternating paths in . (In the rest of this paper, an equivalent statement is always indicated by the same reference number with a prime added.)
Theorem 1 Let and be integers with and , and let be a connected bipartite graph of order with bipartition and be a perfect matching of . If for every two vertices and with , then has an -path-factor with exactly
-alternating paths of order at least .
The proof of Theorem 1 consists of two steps: The first is packing directed paths of a specified order in a digraph, i.e., is to show the existence of vertex-disjoint directed paths of a specified order in a digraph. The second is partitioning a digraph into a specified number of directed paths, i.e., is to extend the collection of directed paths in the first step to a spanning subdigraph. In particular, for simplicity of discussions, we consider the proof of these two steps in bipartite graphs, i.e., we consider the proof of Theorem 1. In Section 2, we first prepare some terminology and notation, and give some lemmas which will be used in the proof of each step. In Section 3, to approach the first step, we consider the problem of giving degree conditions for long directed paths in digraphs (long alternating paths in bipartite graphs), see Theorem 2 (Theorem 2) in Section 3. In Section 4, we prove our main theorem by focusing on the second step in the proof. In Section 5, we give some remarks on the relationships with the problem of giving degree conditions for directed Hamilton cycles with some additional properties.
Section snippets
Preliminaries
In this section, we prepare terminology and notation which will be used in the subsequent sections, and we also give some lemmas.
Let be a graph. We denote by the neighborhood of a vertex in . For and , we let and . For , denotes the subgraph induced by in , and let . We often identify a subgraph of with its vertex set (e.g., we often use instead of for a vertex and a subgraph of ). We
Packing directed paths of a specified order in digraphs
In this section, we give degree conditions for packing directed paths of a specified order in digraphs. For this purpose, we consider the problem of giving degree conditions for the existence of long directed paths in digraphs, since the existence of a directed path of order at least (for given two positive integers and ) implies the existence of vertex-disjoint directed paths of order .
In 1981, Bermond et al. proved the following theorem. Related results can also be found in [10], [11]
Proof of Theorem 1
In this section, we prove Theorem 1. Since Theorems 1 and 1 are equivalent, it suffices to show that Theorem 1 is true.
Before giving the proof of Theorem 1, we need the following additional lemma, which is an improvement of[5, Lemma 4]. The proof is essentially the same as the proof of Theorem 2, and so we give a sketch of the proof.
Lemma 3 Let be a connected bipartite graph of order with a perfect matching . If for every longest -alternating path with endvertices and , we have
Concluding remarks and related problems
In this paper, we have considered degree conditions for the existence of a directed path-factor with a specified number of paths and have proved Theorem 1. In particular, to show this theorem, we have considered two steps: packing directed paths of a specified order and partitioning a digraph into a specified number of paths.
On the other hand, as another approach, it can be considered the problem of giving degree conditions for the existence of a directed Hamilton cycle (with some additional
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the referee for valuable suggestions and comments. The work of S.C. was supported by JSPS, Japan KAKENHI Grant Nos. 17K05347 and 20K03720.
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