The H-force sets of the graphs satisfying the condition of Ore's theorem

Let $G$ be a Hamiltonian graph with $n$ vertices. A nonempty vertex set $X\subseteq V(G)$ is called a Hamiltonian cycle enforcing set (in short, an $H$-force set) of $G$ if every $X$-cycle of $G$ (i.e., a cycle of $G$ containing all vertices of $X$) is a Hamiltonian cycle. For the graph $G$, $h(G)$ is the smallest cardinality of an $H$-force set of $G$ and call it the $H$-force number of $G$. Ore's theorem states that the graph $G$ is Hamiltonian if $d(u)+d(v)\geq n$ for every pair of nonadjacent vertices $u,v$ of $G$. In this paper, we study the $H$-force sets of the graphs satisfying the condition of Ore's theorem, show that the $H$-force number of these graphs is possibly $n$, or $n-2$, or $\frac{n}{2}$ and give a classification of these graphs due to the $H$-force number.


Terminology and introduction
In this paper, we study the simple graphs without loops or no parallel edges. For terminology and notations not defined here we refer the reader to [1]. Let G be a graph with n vertices. The vertex set and the edge set of G are denoted by V (G) and E(G), respectively. For a subset X ⊆ V (G), the cardinality of X is denoted by |X|. A subgraph induced by a subset X is denoted by G[X]. In addition, Let u, v be two distinct vertices and e be an edge of D. We say that a vertex v is incident to an edge e if v is an endpoint of e. If the vertices u, v are incident to the same edge, we say that u and v are adjacent. Let S be a subset of V (G) or a subgraph of G and v is not in S. The set N S (v) = {x| vx ∈ E, x ∈ S} and d S (v) = |N S (v)|. Let δ(G) denote the minimum degree of G.
A graph G is said to be k-connected, if V (G) ≥ k + 1 and G − S is connected for each S ⊆ V (G) with |S| ≤ k − 1. The subset S ⊆ V (G) is called an independent set of G if any pair of vertices in S are nonadjacent. In addition, the subset S is called the largest independent set of G if there is no independent set S ′ such that |S ′ | > |S|.
For two disjoint graphs G and H, G ∨ H is the graph that every vertex of G is adjacent to every vertex of H. A complete graph with n vertices is denoted by K n . K m,n is a complete bipartite graph with two partitions V 1 , V 2 and |V 1 | = m, |V 2 | = n.
A cycle of G is called the Hamiltonian cycle, if it contains all vertices of G. The graph G is said to be a Hamiltonian graph if it has a Hamiltonian cycle. It is well known that the problem to check whether a graph has a Hamiltonian cycle or not is NP-complete. The various kinds of sufficient conditions to imply a graph to be Hamiltonian have been studied widely. More details can be found in [2][3][4][5].
Let G be a Hamiltonian graph with n vertices. For a subset X ⊆ V (G), an X-cycle of G is a cycle containing all vertices of X. A nonempty vertex set X ⊆ V (G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X-cycle of G is a Hamiltonian cycle. For the graph G, h(G) is the smallest cardinality of an H-force set of G and call it the H-force number of G. The subset X is called the minimum H-force set of G if X is an H-force set with |X| = h(G), i.e., X is an H-force set and there is no The H-force set and H-force number were introduced by Fabrici et al. in [6]. These concepts play an important role on the Hamiltonian problems. In [6], the authors also investigated the H-force numbers of the bipartite graphs, the outerplanar Hamiltonian graphs and so on. Recently, these concepts were generalized to digraphs by Zhang et al. in [7] and to hypertournaments by Li et al. in [8], and they gave the characterization of the minimum H-force sets of locally semicomplete diagraphs and hypertournaments, and obtained their H-force numbers.
If d(u) + d(v) ≥ n for every pair of nonadjacent vertices u and v of G, we say that G satisfies the condition of Ore's theorem. For convenience, we call a graph satisfying the condition of Ore's theorem an OTG. In [9], Ore proved that any OTG is Hamiltonian. In this paper, we study the H-force sets and H-force number of the OTG's.
The following is an easy observation on the H-force sets of a graph.
To present Theorem 1.2, we consider a special class of graphs, namely, where K c m is a set of m vertices (also as the complement of the complete graph K m ) and u is another single vertex, furthermore, the edge set of Z m ∨ (K c m + {u}) consists of E(Z m ) and {xy | x ∈ V (Z m ) and y ∈ K c m ∪ {u}}(see Fig.1). In [2], Li et al. investigated the graphs satisfying d(u) + d(v) ≥ n − 1 for every pair of vertices u, v with d(u, v) = 2 and proved the following.
Theorem 1.2 [2] Let G be a 2-connected graph with n ≥ 3 vertices. If d(u)+d(v) ≥ n−1 for every pair of vertices u, v with d(u, v) = 2, then G is a Hamiltonian graph, unless n is odd and G ∈ ψ n .
By Theorem 1.2, we obtain immediately the following corollary.
for every pair of nonadjacent vertices u, v of G, then G is a Hamiltonian graph, unless n is odd and G ∈ ψ n . Proof. Firstly, we prove (1). Note that a cycle of the graph G is always a cycle of the graph G + uv. It is easy to see that if X is an H-force set of G + uv, then X is an H-force set of G.
Suppose that X is an H-force set of G, but not an H-force set of G + uv. Let C be the longest nonhamiltonian cycle containing all vertices of X in G + uv and H be the subgraph induced by V (C) in G, i.e., H = G[V (C)]. Clearly, the cycle C contains the edge uv. Assume that there are a(≥ 1) vertices apart from V (C). Then H contains n − a vertices.
where the subscripts are taken modulo n − a.
Recall that C is the longest nonhamiltonian cycle containing all vertices of X in G+uv. For a > 1, the vertices u and v have no common adjacent vertex apart from V (C). So Fig.2). This contradicts with the fact that X is an H-force set of G. Thus if the subset X is an H-force set of G, X is an H-force set of G + uv.
By (1), it is easy to see that (2) holds.
Lemma 2.1 motivates the following definition. The weak closure of an OTG G is the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n + 1 until no such pair remains. We denote the weak closure of G by C w (G). The idea of weak closure is followed by Bondy and Chvátal's introduction of the closure of a graph. If we join pairs of nonadjacent vertices with the degree sum at least n until no such pair remains, then we obtain the closure of a graph, denoted by C(G). Bondy and Chvátal proved that C(G) is well defined. That means, if G 1 and G 2 are two graphs obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n until no such pair remains, then G 1 = G 2 . By the similar argument, we can see that C w (G) is well defined. Lemma 2.1 implies that the H-force set and H-force number of an OTG G are the H-force set and H-force number of the C w (G), respectively. We can obtain the H-force set and H-force number of G by studying the weak closure C w (G) of G. Clearly, for an OTG G, the weak closure C w (G) is a complete graph or a graph satisfying d Cw(G) (u) + d Cw(G) (v) = n for every pair of nonadjacent vertices u and v. Theorem 2.2 Let G be an OTG with n ≥ 5 vertices and X be the minimum H-force set of G. Let C w (G) be the weak closure of G and S be the largest independent set of C w (G).

Then
(1) the H-force number h(G) = n − 2, n 2 , or n, and (2) Proof. According to Lemma 2.1, it is sufficient to consider the minimum H-force set and H-force number of the weak closure C w (G). For the convenience, let So assume that G w is not a complete graph and satisfies d Gw (u) + d Gw (v) = n for every pair of nonadjacent vertices u and v in G w . Clearly, 2 ≤ δ(G w ) ≤ n 2 . We consider the following two cases.
(ii) W induces a complete subgraph.
Let w ∈ W , and x, y ∈ U be the vertices adjacent to w in U.  Proof. (i) For the vertex w ∈ W , since w is not adjacent to u, we see that d Gw (w) = n − d Gw (u) = n − a.
(ii) For the distinct vertices w, w ′ ∈ W , we have d Gw (w) = d Gw (w ′ ) = n − a > n 2 , and hence d Gw (w) + d Gw (w ′ ) > n. So w and w ′ are adjacent. By the choice of w, w ′ , W induces a complete subgraph.
(iii) Since |W | = n − a − 1 and W is a complete subgraph, we have (iv) Since the vertex p is not adjacent to w and d Gw (w) = n − a, we have d Gw (p) = n − d Gw (w) = n − (n − a) = a.
( (vii) For q ∈ U − {x, y}, the claims (iv), (v) and (vi) imply that q is not adjacent to any vertex of W .
(viii) The claims (iii) and (vii) show that every vertex of W has two adjacent vertices in U, which are just x and y.
We shall show that X = V − {x, y} = U ′ ∪ W is the minimum H-force set and hence h(G w ) = n − 2.
Obviously, X is an H-force set because a cycle containing all vertices of X must encounter x and y. Suppose to the contrary that X ′ is an H-force set of G w with |X ′ | < |X|. Clearly, there exists a vertex z ∈ U ′ or z ∈ W such that z / ∈ X ′ . Without loss of generality, assume z ∈ U ′ . Consider the subgraph G w − z. For any pair of nonadjacent Then G w − z is an OTG and hence G w − z is a Hamiltonian graph. In other words, there is a nonhamiltonian cycle containing all vertices of X ′ in G w , a contradiction. Thus there doesn't exist such an H-force set X ′ with |X ′ | < |X|. So X is the minimum H-force set and h(G w ) = h(G) = n − 2.
Case 2: δ(G w ) = n 2 and n is even.
For arbitrary x, y ∈ V (H), we have By Corollary 1.3, either H is not a 2-connected graph or H ∈ ψ n−1 (see Fig.1).
This implies that U induces a complete subgraph and x is adjacent to u, v. Similarly, W induces a complete subgraph and every vertex of W is adjacent to u, v.
Furthermore, all vertices of U and W are the neighbours of u and v. So d Gw (u) ≥ n−2, d Gw (v) ≥ n − 2 and hence d Gw (u) + d Gw (v) > n. It means that u and v are adjacent. Figure 4: G 12 ∈ ϕ 1 In subcase 2.1.1, G w ∼ = G 12 (see Fig.4), in which U and W are complete subgraphs with |U| = |W | = n 2 − 1. By the same argument of the case when 2 ≤ δ(G w ) < n 2 and G w ∼ = G 11 , we also have V −{u, v} is the minimum H-force set and h(G w ) = h(G) = n−2.
Claim C: If there is at least one edge in Z m , then (i) Z m is a complete subgraph.
(ii) The vertex v is adjacent to all vertices of Z m .
(iii) The vertex v is adjacent to all vertices of K c m+1 . (iv) G w ∼ = G 21 = K c m+1 ∨ K m+1 (see Fig.5). Furthermore, X = V (K c m+1 ) is the minimum H-force set and h(G) = h(G w ) = n 2 . Proof. (i) Let x be an endpoint of the edge of Z m . Then d Gw (x) ≥ m + 2. For arbitrary y ∈ Z m −x, since d Gw (y) ≥ m+1, we have d Gw (x)+d Gw (y) ≥ 2m+3 > n. This implies that x is adjacent to the remaining vertices of Z m . Furthermore, for any pair of p, q ∈ Z m − x, we have d Gw (p) ≥ m + 2, d Gw (q) ≥ m + 2, and hence d Gw (p) + d Gw (q) ≥ 2m + 4 > n, which implies that p and q are adjacent. Thus Z m is a complete subgraph.
(ii) For a vertex z ∈ Z m , Thus v is adjacent to all vertices of Z m . Figure 5: G 21 ∈ ϕ 2 and K n 2 , n 2 ∈ ϕ 2 (iv) By the arguments above, it is obvious For any x ∈ X, it is obvious that G w − x is an OTG and hence it is Hamiltonian. By Proposition 1.1, the minimum H-force set contains all vertices of V (K c m+1 ). In addition, it is clear that In [10], Dirac proved that Theorem 2.3.

Theorem 2.3 [10]
If the minimum degree δ(G) of G is no less than n 2 , then G is Hamiltonian.
Clearly, the graph satisfying the condition of Dirac's theorem is an OTG. By Theorem 2.2, we obtain immediately the following results on the H-force sets of graphs satisfying the condition of Dirac's theorem.
Corollary 2.4 Let G be a graph satisfying δ(G) ≥ n 2 and X be the minimum H-force set of G. Let G w = C w (G) be the weak closure of G and S be the largest independent set of G w . Then (1) The H-force number h(G) = n − 2, n 2 , or n.

The classification of the OTG's
To present Theorem 3.1, we define several special classes of graphs, namely, Fig.3 and Fig.4),  Let C w be the class of the weak closures of all OTG's. Clearly, ϕ 1 , ϕ 2 , ϕ 3 are all the subsets of C w . By Theorem 2.2, we can give a partition of the class C w and obtain the following theorem.
due to the H-force set, where ϕ 1 , ϕ 2 , ϕ 3 are defined above. Let G w ∈ C w be arbitrary. Then (1) Proof. By the case 1 and subcase 2.1.1 of Theorem 2.2, the statement (1) holds. Also by the subcase 2.1.2, the statement (2) holds. We show (3) as follows.
"=⇒" By the proof of Theorem 2.2, we know that h(G w ) = n − 2 if 2 ≤ δ(G w ) < n 2 . So if h(G w ) = n, then δ(G w ) = n 2 (n is even) or G w is a complete graph. In the latter case, G w ∈ ϕ 3 and we are done.
Assume that G w is not a complete graph. Then δ(G w ) = n 2 and d Gw (u) + d Gw (v) = n for every pair of nonadjacent vertices u, v. So the degree of any vertex u ∈ V (G w ) is either n 2 or n − 1 . Set U = {v| d Gw (v) = n − 1} and m = |U|. When U = ∅, we claim that 1 ≤ m < n 2 . Clearly, 1 ≤ m ≤ n 2 . Note that d Gw (w) = n 2 for any vertex w ∈ V (G w )−U. If m = n 2 , then G w ∼ = G 21 and h(G w ) = n 2 , a contradiction. So 1 ≤ m < n 2 . Since h(G 12 ) = n − 2, we have G w ≇ G 12 . Thus, G w ∈ ϕ 3 . When U is an empty set, then m = 0 and G w is a n 2 -regular graph. Since h(K n 2 , n 2 ) = n 2 , we have G w ≇ K n 2 , n 2 . Thus, G w ∈ ϕ 3 . "⇐=" It is not difficult to check that ϕ 3 ∩ ϕ 1 = ∅ and ϕ 3 ∩ ϕ 2 = ∅. Let G w ∈ ϕ 3 be a weak closure of an OTG. So h(G w ) = n − 2 or n 2 . According to Theorem 2.2, h(G w ) = n.
Let ϑ be the class of all OTG's. Clearly, ϑ 1 , ϑ 2 , ϑ 3 are all the subsets of ϑ. By Theorem 3.1, we can give a partition of the class ϑ and obtain the following Corollary.