On vertex-disjoint paths in regular graphs

Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\lfloor 1/c \rfloor$ is best possible when $1/c\notin \mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\lfloor 1/(2c) \rfloor$, which is best possible.


Introduction
Paths and cycles are fundamental objects in graph theory. The path cover number is the minimum number of vertex-disjoint paths whose union covers all vertices of G. Note that we allow paths of length 0 (single vertices) in the definition above. Trivially the path cover number of a graph G is upper bounded by the independence number of G, because the set of the (arbitrary one out of the two) end vertices of the paths in a minimal path cover form an independent set in G. It is evident that for general graphs, determining the path cover number is NP-hard, because deciding if the path cover number equals 1 is equivalent to the decision problem for Hamiltonian path, which is NP-complete. For bounds on the path cover number for general graphs, see e.g. [3,6]. For regular graphs, Magnant and Martin [5] made the following conjecture and confirmed it for 0 ≤ k ≤ 5. Conjecture 1.1. If G be a k-regular graph of order n, then the path cover number of G is at most n/(k + 1).
If true, the bound in Conjecture 1.1 would be tight as seen by disjoint copies of complete graph K k+1 (if n ≡ j modulo k + 1 and j = 0, then we change one copy of K k+1 to a copy of K k+1+j ). By the celebrated Dirac theorem on Hamiltonian paths [2], Conjecture 1.1 is true for k ≥ (n − 1)/2. As far as we know, Conjecture 1.1 is open for all other cases. To provide more evidence on the validity of the conjecture, in this note we prove the following result for dense regular graphs. Theorem 1.2. For any c, α > 0, there exists n 0 ∈ N such that the following holds for every integer n ≥ n 0 .
(1) Every ⌈cn⌉-regular graph of order n contains a collection of at most ⌊1/c⌋ vertex-disjoint paths whose union covers all but αn vertices. (2) Every bipartite ⌈cn⌉-regular graph of order n contains a collection of at most ⌊1/(2c)⌋ vertexdisjoint paths whose union covers all but αn vertices.
Note that Part (2) of the theorem corresponds to the bipartite version of Conjecture 1.1: if G is a bipartite k-regular graph of order n, then the path cover number of G can be as large as n/(2k), as seen by the vertex-disjoint copies of K k,k . Note that So when n is large, if 1/c / ∈ N, then ⌊ n ⌈cn⌉+1 ⌋ = ⌊1/c⌋, i.e., the number of paths in Theorem 1.2 matches the quantity in Conjecture 1.1; however if 1/c ∈ N, then the quantity ⌊1/c⌋ is off by 1. On the other hand, the quantity ⌊1/(2c)⌋ in Part (2) is optimal.
At last, we remark that the bound in Conjecture 1.1 is not tight if we restrict the problem on connected regular graphs, see [7] for connected cubic graphs.

A weaker result
We first prove the following weaker result. For reals a, b, c, we write a = (1 ± b)c if there exists a real x ∈ (1 − b, 1 + b) such that a = xc. Theorem 2.1. Given any reals c, α > 0, there exists ǫ > 0 and integer C > 0 such that the following holds for sufficiently large integer n. Let G be a graph of order n such that deg(v) = (1 ± ǫ)cn for every v ∈ V (G). Then there exists a collection of C vertex-disjoint cycles in G whose union covers all but αn vertices of G.
Our main tools for embedding the cycles are the Regularity Lemma of Szemerédi [11] and the Blow-up Lemma of Komlós et al. [4]. For any two disjoint vertex-sets A and B of a graph G, the density of A and B is defined as d Lemma 2.2 (Regularity Lemma -Degree Form). For every ǫ > 0 there is an M = M (ǫ) such that for any graph G = (V, E) and any real number d ∈ [0, 1], there is a partition of the vertex set V into t + 1 clusters V 0 , V 1 , . . . , V t , and there is a subgraph G ′ of G with the following properties: The Blow-up Lemma allows us to regard a super regular pair as a complete bipartite graph when embedding a graph with bounded degree. Since we will always use it to embed a cycle, we state it in the following special form. Proof of Theorem 2.1. Given c, α > 0, let d = αc/9. We apply Lemma 2.3 with δ = d/2 and obtain ǫ 1 > 0. Let ǫ = min{ǫ 1 , d/6, 3d/(2c)}. We then apply Lemma 2.2 with ǫ and obtain M = M (ǫ). Let n ∈ N be sufficiently large. Let G = (V, E) be a graph of order n such that deg(v) = (1 ± ǫ)cn for every v ∈ V . We apply the Regularity Lemma (Lemma 2.2) on G with the constants ǫ, d chosen as above and obtain a partition of V into V 0 , V 1 , . . . , V t for some t ≤ M , and a subgraph G ′ of G with the properties as described in Lemma 2.2. By moving at most one vertex from each V i , i ∈ [t] to V 0 , we can assume that m := |V i | is even. Thus we have |V 0 | ≤ ǫn + t ≤ 2ǫn. Now for any v ∈ V , Let β = 3d/c. Let H be the graph on [t] such that ij ∈ E(H) if and only if d(V i , V j ) ≥ d. We first assume that there exists a set S ⊆ [t], such that i(H − S) − |S| ≥ βt. In particular, let T be the collection of |S| + βt isolated vertices in H − S. Thus we have However, by averaging, this implies that there exists a vertex v ∈ V such that For each i ∈ [t] we arbitrarily split V i into V 1 i and V 2 i each of size m/2. Thus the existence of f implies that we can partition V \ V 0 into at least (1 − β)t/2 pairs of sets each of form (V a i , V b j ) with density at least d, where i, j ∈ [t], i = j and a, b ∈ [2], and a set of at most 2βt · m vertices. Note that here (to simplify the argument) even if an edge ij ∈ E(H) receives weight 1, we still split it, e.g., as (V 1 i , V 1 j ) and (V 2 i , V 2 j ). Thus every vertex of H is in at most two pairs so there are at most t pairs. We will show that each such pair contains a cycle that covers all but at most 2ǫm vertices. Indeed, fix any pair (V a i , V b j ), let A be the set of vertices in V a i whose degree to V b j is less than Similarly let B be the set of vertices in V b j whose degree to V a i is less than (d − ǫ)|V a i | and we have |B| ≤ ǫm. Let A ′ ⊇ A and B ′ ⊇ B be arbitrary subsets of V a i and V b j , respectively, of size exactly ǫm. Now let -super-regular with density at least d − 3ǫ, and |X| = |Y | = m − ǫm. Since d − 3ǫ ≥ d/2, by Lemma 2.3, (X, Y ) contains a spanning cycle and we are done.
Let C = M . Thus we obtain a set of at most t ≤ M = C vertex-disjoint cycles in G that covers all but at most t · 2ǫm + |V 0 | + 2βt · m ≤ 3βn = αn vertices, completing the proof.

Proof of Theorem 1.2
In the proof of Theorem 1.2 we use the trick of a 'reservoir lemma' from [8,9]. Roughly speaking, we will reserve a random set R of vertices at the beginning of the proof, and use them to connect the paths returned by applying Theorem 2.1 on G − R. We first recall the following Chernoff's bounds (see, e.g., [1]) for binomial random variables and for x > 0: Lemma 3.1. Given any c, γ, ǫ > 0, the following holds for sufficiently large integer n. Let G be a ⌈cn⌉regular graph of order n. Then there exists a set R ⊆ V (G) such that |R| = (1 ± ǫ)γn and every vertex of G has degree (1 ± ǫ)cγn in R.
Next we iteratively use the property of R to connect some pair of paths. We first explain the general case. Suppose there are at least ⌊1/c⌋ + 1 paths left. Indeed, let v 1 , v 2 , . . . , v ⌊1/c⌋+1 be the (arbitrary one out of the two) ends of the ⌊1/c⌋ + 1 paths. Note that throughout the iteration there are at most C vertices in R that have been used for connecting and thus removed from R. So for each i, by by the definition of ǫ. Thus there exist two vertices v i , v j which have a common neighbor w in R so that we can connect the corresponding two paths by w. At the end, we obtain a collection of at most ⌊1/c⌋ vertex-disjoint paths whose union covers all but at most αn/2 + (1 + ǫ)γn ≤ αn vertices in G.
Second, assume that there are at least ⌊1/(2c)⌋ + 1 paths left and in addition that G is bipartite with bipartition X and Y . Note that since G is regular we have |X| = |Y | = n/2 (so in particular n must be even). Fix ⌊1/(2c)⌋ + 1 paths. By throwing away at most one vertex from each path we can assume that each path has exactly one end vertex in X and one in Y . Let v 1 , v 2 , . . . , v ⌊1/(2c)⌋+1 be the end vertices in X. By the similar calculation, we can find a vertex w ∈ R ∩ Y which connects some pair of paths. At the end, we obtain a collection of at most ⌊1/(2c)⌋ vertex-disjoint paths whose union covers all but at most αn/2 + (1 + ǫ)γn + C 2 ≤ αn vertices in G, because the iteration has at most C steps and in each step we threw away at most C vertices from the current paths.