On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

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Introduction
In this paper, all graphs are undirected, simple, and without loops.For graph definitions and notations, we refer to Hsu and Lin (2009).We denote any graph by G = (V, E), where V is the vertex set and E ⊆ {(u, v) | (u, v) is an unordered pair of V } the edge set of G.The order of a graph G, denoted by |G|, is the number of vertices of G.An edge of G is denoted by (u, v), where u, v ∈ V and (u, v) ∈ E. Two vertices u and v are adjacent in G if there is an edge (u, v) in G.The degree of a vertex u in G, denoted by deg G (u), is the number of vertices adjacent to u.The notation δ(G) represents the minimum degree of vertices of the graph G.A walk of length k is denoted by v 0 , v 1 , . . ., v k , where v i 's are vertices such that (v i−1 , v i ) ∈ E for all i.A walk from u to v starts from the first vertex u and ends at the last vertex v; u and v are called the endvertices.A path is a walk with no repeated vertex.A graph G is traceable if G contains a Hamiltonian path.A cycle is a closed walk in which the first vertex and the last vertex are the only vertex repetition.A Hamiltonian cycle of G is a cycle that traverses every vertex of G exactly once.A Hamiltonian graph is a graph with a Hamiltonian cycle.A graph G is connected if there is a path between any two distinct vertices in G and is Hamiltonian-connected if there is a Hamiltonian path between any two distinct vertices in G.The symbol G − F , F ⊆ V , denotes the graph with V (G − F ) = V − F , and E(G − F ) obtained by E after deleting the edges with at least one endvertex in F .A graph G is k-vertex fault traceable (resp.k-vertex fault Hamiltonian, k-vertex fault Hamiltonian-connected) if G − F remains traceable (resp.Hamiltonian, Hamiltonian-connected) for any set Kao et al. (2006).
We use a b to denote the binomial coefficient indexed by a and b, where a and b are positive integers and a ≥ b.Let G 1 and G 2 be two graphs.We say that G 1 and G 2 are disjoint if G 1 and G 2 have no vertex in common.The union of two disjoint graphs G 1 and G 2 , denoted by to each vertex of G 2 with an edge.We use K n for a complete graph with n vertices and K n for the union of n isolated vertices.
If G is a graph with |G| = n and degrees and the nondecreasing degree sequence of G is majorized by that of H.For example, the 5-cycle is degree-majorized by the complete bipartite graph K 2,3 since (2, 2, 2, 2, 2) is majorized by (2, 2, 2, 3, 3).
Ever since Dirac's theorem for hamiltonicity was established, theorems for various Hamiltonian properties have been derived based on the degree conditions of a given graph.Some of the well-known results are presented below.
Theorem 1 (Ore (1960(Ore ( , 1963))) Let G = (V, E) be a graph with Theorem 2 (Chvátal (1972)) Let G be a graph with Obviously, compared with degree sums, the concept of degree sequences characterizes a graph in a more refined way.Consider G i = (2K 2 + K 1 ) ∨ K i+1 , where 1 ≤ i ≤ 2, for example.It is easy to see that |G i | = 6 + i. Applying Theorem 2(i) on G 1 and (ii) on G 2 , we know that G 1 is traceable and G 2 is Hamiltonian.However, for any pair of non-adjacent vertices {u, v} in G i , G 1 fails to satisfy the condition Let G 3 be obtained as (K 2 + K 4 ) ∨ K 3 with an additional edge between a vertex of K 2 and K 4 .Then G 3 is a Degree-sequence and number of edges for graphs with Hamiltonian properties under fault tolerance 309 Hamiltonian-connected graph, which satisfies Theorem 3 but fails to satisfy Theorem 1.These examples show that the degree sequence of a graph helps to determine the associated Hamiltonian properties when its degree sum condition (as in Theorem 1) gives no conclusion.
In the past decade, some results regarding the minimum number of edges that guarantees various properties have been published.In Brandt (1997), Bollobás andThomason (1999), andErdős et al. (1996), for example, the minimum number of edges is given as a function of the total number of vertices of any graph.In Ho et al. (2010Ho et al. ( , 2011)), Ho and his coauthors studied the minimum number of edges required to guarantee an n-vertex graph G with minimum degree δ(G) ≥ δ to be Hamiltonian or Hamiltonian-connected, and expressed it as a function of |G| = n and the minimum degree δ(G) ≥ δ.Such results have many applications in interconnection networks under conditional faults, and provide better lower bounds for the number of edges by taking δ into account.See Ho et al. (2010Ho et al. ( , 2011) ) and their references.Our present results extend the formulas of Ho et al. Chvátal (1972) characterized the degree sequence behavior for a graph to remain Hamiltonian after the removal of up to k faulty vertices.To our knowledge, other than this study, no result about the vertex fault version of Theorem 2 and 3 has been published.Inspired by the above-mentioned works, we intend to establish two main theorems as follows.Note that Theorem 4(ii) was proved by Chvátal (1972).
(ii) Let k be an integer with 0 In the sequel, the notation a mod b denotes the remainder of the division of a by b.
Theorem 5 Assume that n, δ, and k are integers with and , and h 3 (n, δ, k) the minimum number of edges required to guarantee the n-vertex graph G with minimum degree δ(G) ≥ δ to be k-vertex fault traceable, k-vertex fault Hamiltonian, and k-vertex fault Hamiltonian-connected, respectively.Then It can be observed that Theorem 2 and Theorem 3 become special cases for Theorem 4 with k = 0; the formula in Ho et al. (2010Ho et al. ( , 2011) ) becomes a special case of Theorem 5 by taking k = 0.In other words, Theorem 5 further extends Ho's formulas for fault tolerant Hamiltonian graphs.
2 Proof of the main theorems Let n be the total number of vertices in a graph and k be an integer with k ≥ 0. We first define three graph families as follows. (1)

Proof of Theorem 4
To prove Theorem 4, we recall the following theorem by Chvátal (1972) first, which contains (ii) of Theorem 4. Note that any graph in the family C k m,n , which was introduced by Chvátal (1972), has the greatest degree sequence among all graphs with the same number of vertices and being not k-vertex fault Degree-sequence and number of edges for graphs with Hamiltonian properties under fault tolerance 311 Theorem 6 (Chvátal (1972)) Let k be an integer with 0 ≤ k ≤ n−3.If the degree sequence (d 1 , d 2 , . . ., d n ) of a graph G satisfies then G is k-vertex fault Hamiltonian.On the other hand, if the degree sequence of G fails to satisfy (4), then it is majorized by the degree sequence of the graph C k m,n , which is not k-vertex fault Hamiltonian.
Proof of Theorem 4: (ii) is directly derived from Theorem 6.We will show (iii) and (i) in order using the result of (ii).
To show (iii), we assume that the sequence (d 1 , d 2 , . . ., d n ) satisfies (3).Let k be an arbitrary integer with 0 ≤ k ≤ k.For any faulty vertex set F = {y 1 , y 2 , . . ., y k } in G, we want to show that for any pair of vertices {u, v} in G − F , there exists a Hamiltonian path between u and v in 3), we have Note that n and k are integers.Thus ( 5) is equivalent to Note that d * j = d j−1 + 1.If d * j > j + k , we do not need to check the above condition.If 6), we have Therefore, according to (7), the degree sequence (d * 1 , d * 2 , . . ., d * n+1 ), with n + 1 in place of n, satisfies (2).By (ii), G * − F is Hamiltonian.Since G * − F has a Hamiltonian cycle if and only if G − F has a Hamiltonian path between u and v, G − F is Hamiltonian-connected.
To show (i), we assume that the sequence (d 1 , d 2 , . . ., d n ) satisfies (1).Let k be an arbitrary integer with 0 ≤ k ≤ k.For any faulty vertex set F = {v 1 , v 2 , . . ., v k } in G, we want to show there exists a Hamiltonian path in G − F .Let G be a graph by adding to G a new vertex x and new edges joining x to all the vertices of G. Let ( d 1 , d 2 , . . ., d n+1 ) be the degree sequence of G.Note that As in (iii), it is easy to show that the degree sequence Proof: We show (iii) only.The proofs for (i) and (ii) can be derived similarly.
According to Theorem 4(iii), if G − F is not k-vertex fault Hamiltonian-connected, then for some k The greatest degree sequence satisfying ( 8) is of the following form: For any graph being not k-vertex fault Hamiltonian-connected, its degree sequence must be degreemajorized by It is easy to see that H k m,n 's degree sequence is the same as above.Consequently, adding an extra edge to H k m,n results in a graph where no index m in (8) exists, which means the sufficient condition of Theorem 4(iii) is satisfied.Therefore, H k m,n + e is k-vertex fault Hamiltonian-connected for any extra edge e. 2

Proof of Theorem 5
Theorem 5 consists of three results.We shall derive h 3 (n, δ, k) in this section.The values h 1 (n, δ, k) and h 2 (n, δ, k) can be obtained by the similar derivations.
Degree-sequence and number of edges for graphs with Hamiltonian properties under fault tolerance 313 is a quadratic function of m and its maximum value occurs at the boundary m Proof: See (9) for |E(H k t,n )|.We finish this proof by the following two cases.

Fig. 1 :
Fig. 1: (a)H k m,n ; (b)H k m,n with a different layout.We will use these three graph families to establish the sharpness of the bounds in Theorem 5. Let the faulty vertex set F ⊆ V (K m+k ) with |F | = k.For example, F = {v n−k−m+i | 1 ≤ i ≤ k} as labeled in Figure 1 for H k m,n .Then it is easy to check that T k m,n − F has no Hamiltonian path, C k m,n − F has no Hamiltonian cycle, and H k m,n − F has no Hamiltonian path between some pair of distinct vertices {u, v} in K m+k − F .Thus we have the following lemma.Lemma 1 (i) Let n ≥ 4, 0 ≤ k ≤ n − 2, and 0 ≤ m ≤ n−k−2

Theorem 7 (
Ore (1963)) Let n ≥ 3. Any simple graph G, where |G| = n, with δ(G) ≥ n+1 2 is Hamiltonian-connected.Corollary 2 Let n and k be integers with n ≥ 4 and 0 ≤ k ≤ n − 4. If G is a graph with |G| = n and δ(G) > n+k 2 , then G is k-vertex fault Hamiltonian-connected.Lemma 2 Let n, m, k be integers with n ≥ 6 and 0 ≤ k ≤ n − 4. Let G be a graph with |G| = n and δ be an integer with k