Asymptotically sharpening the $s$-Hamiltonian index bound

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $\ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. \begin{equation*} h_s(G)\le \left\{ \begin{aligned}&\ell(G)+1,&&\mbox{ if }\delta(G)\le 2 \mbox{ and }s=0;\\&\widetilde d(G)+2+\lceil \lg (s+1)\rceil,&&\mbox{ if }\delta(G)\le 2 \mbox{ and }s\ge 1;\\&2+\left\lceil\lg\frac{s+1}{\delta(G)-2}\right\rceil,&&\mbox{ if } 3\le\delta(G)\le s+2;\\&2,&&{\rm otherwise}, \end{aligned} \right. \end{equation*} where $\widetilde d(G)$ is the smallest integer $i$ such that $\delta(L^i(G))\ge 3$. Consequently, when $s \ge 6$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(\ell(G)+s+1)$ as $s \to \infty$. This sharpens the result, $h_s(G)\le\ell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].


Introduction
Finite loopless graphs permitting parallel edges are considered with undefined terms being referenced to [5]. As in [5], a simple graph is one that is loopless and without parallel edges, and the minimum degree of a graph G is denoted by δ(G). For a subset X ⊆ V (G) or E(G), let G[X] denote the subgraph induced by X, and let G − X = G[V (G) − X] or G[E(G) − X], respectively. When X = {x}, we write G − x for G − {x}. Throughout this paper, if X ⊆ E(G), then, for notational convenience, we often use X to denote both the edge subset of E(G) and G[X]. We also use lg x as an alternative notation for log 2 x, and set [m, n] = {m, m + 1, . . . , n} for two integers m, n with m ≤ n.
A graph is considered Hamiltonian if it has a spanning cycle. For a non-negative integer s ≤ |V (G)| − 3, a graph is called s-Hamiltonian if the removal of any k ≤ s vertices results in a Hamiltonian graph. A subgraph H of G is dominating if G − V (H) is edgeless.
Following [4,6], a graph is supereulerian if it has a spanning closed trail. Harary and Nash-Williams [13] characterized Hamiltonian line graphs as follows, which implies that the line graph of every supereulerian graph is Hamiltonian. Theorem 1.1 (Harary and Nash-Williams, Proposition 8 of [13]). Let G be a graph with at least three edges. Then L(G) is Hamiltonian if and only if G has a dominating closed trail.
The line graph of a graph G, denoted L(G), is a simple graph with E(G) being its vertex set, where two vertices in L(G) are adjacent whenever the corresponding edges in G are adjacent. A claw-free graph is one that does not have an induced subgraph isomorphic to K 1,3 . Beineke [2] and Robertson (Page 74 of [12]) showed that line graphs are claw-free graphs. For a positive integer i, we define L 0 (G) = G, and the ith iterated line graph of G, denoted L i (G), is defined recursively as L i (G) = L(L i−1 (G)).
Let J 1 and J 2 be two graphs obtained from K 1,3 via identifying two and three vertices of degree 1, respectively. Let K + 1,3 = {J 1 , J 2 , K 1,3 }. Since the line graph of a cycle remains unchanged, in general, we assume that graphs are not isomorphic to paths, cycles or any members in K + 1,3 . For this reason, we define G = {G : G is connected and is not isomorphic to a path, or a cycle, or a member in K + 1,3 }.
Chartrand in [9] introduced and studied the Hamiltonian index of a graph, and initiated the study of indices of graphical properties. More generally, we have the following definition.
Definition 1.1 (Definition 5.8 of [17]). For a property P, the P-index of G ∈ G is defined by A property P is line graph stable if L(G) has P whenever G has P. Chartrand [9] showed that for every graph G ∈ G, the Hamiltonian index exists as a finite number, and the characterization of Hamiltonian line graphs (Theorem 1.1) by Harary and Nash-Williams implies that being Hamiltonian is line graph stable. Z. Ryjáček et al. [25] indicated that determining the value of the Hamiltonian index is difficult. Clark and Wormald [11] showed that for all graphs in G, other Hamiltonian-like indices also exist as finite numbers; and in [17], it is shown that these Hamiltonian-like properties are also line graph stable. Let h(G), h s (G) and s(G) be the Hamiltonian index, s-Hamiltonian index and supereulerian index of G ∈ G, respectively. By definitions, h(G) = h 0 (G).
Let P = v 0 e 1 v 1 e 2 · · · v s−1 e s v s be a path of a graph G where each e i ∈ E(G) and each v i ∈ V (G). Then P is called a (v 0 , v l )-path or an (e 1 , e s )-path of G. A path P of G is divalent if every internal vertex of P has degree 2 in G. For two non-negative integers s and t, a divalent path P of G is a divalent (s, t)-path if the two end vertices of P have degrees s and t, respectively. A non-closed divalent path P is considered proper if P is not both of length 2 and in a K 3 . As in [16,26], for a graph G ∈ G, define ℓ(G) = max{m : G has a length m proper divalent path}.
(1) Several natural questions arise here. Can we improve the upper bounds above? Can we generalize Theorem 1.2(i) in the way as Theorem 1.2(iii) extends Theorem 1.2(ii)? As a generalization of supereulerian graphs, given two non-negative integers s and t, it is defined in [21] that a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X| ≤ s and |Y | ≤ t, G − Y contains a spanning closed trail that traverses all edges in X. Former studies on (s, t)-supereulerian graphs can be found in [19][20][21], among others. Let i s,t (G) denote the (s, t)-supereulerian index of a graph G ∈ G. Thus, i 0,0 (G) = s(G). By the characterization of Hamiltonian line graphs (Theorem 1.1), the line graph of every (0, s)-supereulerian graph is s-Hamiltonian, and then we obtain the following observation.
To present the main results, an additional notation would be needed. Since G ∈ G, it is observed that (for example, Theorem 18 of [10]) there exists an integer i > 0 such that Our main results can now be stated as follows.
. Then, given two non-negative integers s and t, Using Observation 1, Theorem 1.3 implies Corollary 1.4 below.
Given a simple graph G ∈ G with ℓ = ℓ(G) and d = d(G). By the formula to compute d to be presented in Section 3.1, we have d ≤ ℓ + 2. When s ≥ 6, as ⌈lg(s + 1)⌉ + 2 ≤ s − 1, In the next section, we present preliminaries and tools that will be used in our discussions. In Section 3, we shall show some important lemmas, including a corrected formula to compute d(G), which are very helpful to prove the main result, Theorem 1.3, in Section 4.

Preliminaries
For a vertex v ∈ V (G), we denote N G (v) to be the set of all neighbors of vertex v in a graph G, and denote E G (v) to be the set of all edges incident with v in G.

Iterated Line Graphs
and v ∈ D t (L(J)). As Q is not in a K 3 and the definition of divalent paths, a divalent (s, t)-path of length r + j 0 , which contradicts our choice of j 0 .

Collapsible Graphs
In [7], Catlin defined collapsible graphs as a useful tool to study supereulerian graphs. A graph G is collapsible if for every subset R ⊆ V (G) with |R| ≡ 0 (mod 2), G has a subgraph Γ R such that O(Γ R ) = R and G − E(Γ R ) is connected. By definition, all complete graphs K n except K 2 are collapsible. As shown in Proposition 1 of [18], a graph G is collapsible if and only if for every subset R ⊆ V (G) with |R| ≡ 0 (mod 2), G has a spanning connected subgraph L R with O(L R ) = R. As L ∅ is a spanning eulerian subgraph, every collapsible graph is supereulerian. Collapsible graphs have been considered to be very useful to study eulerian subgraphs via the graph contraction. For an edge subset X ⊆ E(G), the contraction G/X is obtained from G by identifying the two ends of each edge in X and deleting the resulting loops. If H is a subgraph of G, then we write G/H for G/E(H). The following theorem summarizes some useful properties of collapsible graphs for our proofs.

The k-Triangular Index
A cycle of length 3 is often called a triangle. Following [3], for an integer k > 0, a graph G is k-triangular if every edge lies in at least k distinct triangles in G; a graph G is triangular if G is 1-triangular. Let T k denote the family of all k-triangular graphs. Thus, δ(G) ≥ k+1 if G ∈ T k . Triangular graphs are often considered as models for some kinds of cellular networks ( [14]) and for certain social networks ( [22]), as well as mechanisms to study network stabilities and to classify spam websites ( [1]). In addition to its applications in the hamiltonicity of line graphs ( [3]), triangular graphs are also related to design theory. In 1984, Moon in [23] introduced the Johnson graphs J(n, s), named after Selmer M. Johnson for the closely related Johnson scheme. The vertex set of J(n, s) is all s-element subsets of an n-element set, where two vertices are adjacent whenever the intersection of the corresponding two subsets contains exactly s − 1 elements. For example, J(n, 1) is isomorphic to K n . By definitions, for any integers n ≥ 3 and s with n > s, J(n, s) is (n − 2)triangular. Therefore, it is of interests to investigate k-triangular graphs for a generic value of k.
For an integer k > 0, define t k (G) to be the k-triangular index of G ∈ G, that is, the smallest integer m such that L m (G) ∈ T k . The triangular index t 1 (G) is first investigated by Zhang et al. One of the purposes of this section is to determine, for any positive integer k, the best possible bounds for t k (G) and to investigate whether being k-triangular is line graph stable.

A Formula to Compute d(G)
Recall that d(G) = min{i : δ(L i (G)) ≥ 3}, which is defined in (2). Define and In [15], it is claimed that "It is easy to see d(G) = ℓ 0 (G)." However, there exists an infinite family of graphs each of which shows that this claim might be incorrect. Let B = {T : T is a tree with V (T ) = D 1 (T ) ∪ D 3 (T )}. For each G ∈ B, we have ℓ 1 (G) = ℓ 3 (G) = 1 and ℓ 2 (G) = 0. Direct computation indicates that d(G) = 3 > ℓ 0 (G). See Figure 1 for an illustration.
G L(G) L 2 (G) Fig. 1: A member G ∈ B and its iterated line graphs.
Thus what would be the correct formula to compute d(G) becomes a question to be answered. Before presenting our answer to it, we need some notation.
Lemma 3.2. Let G ∈ G be a graph with δ(G) ≤ 2, d = d(G) and ℓ 0 = ℓ 0 (G). The formula below computes d: Proof: Let m be the right-hand side of (6) and let ℓ i = ℓ i (G) for each i ∈ {1, 2, 3}. Then m ≤ d by definitions of d and line graphs. Now, it suffices to show that δ(L m (G)) ≥ 3. We assume that δ(L m (G)) ≤ 2 to seek a contradiction.
Then, δ(L m (G)) = 2. Pick u ∈ D 2 (L m (G)). If u is not in any triangles of L m (G), then u is in a divalent (s ′ , t ′ )-path of length r ′ ≥ 2 in L m (G) that is not in a K 3 , where s ′ ≥ 3 and t ′ ≥ 3. It follows that G has a divalent (s ′ , t ′ )-path of length r ′ + m by Lemma 2.1, which shows that 2 + m ≤ r ′ + m ≤ ℓ 3 ≤ m + 1, a contradiction. Thus, u ∈ V (H) where H ∼ = K 3 is a subgraph of L m (G). By the definition of line graphs, L −1 (H) is isomorphic to one member of {K 3 , K 1,3 , J 1 , J 2 }. Let u = xy ∈ E(L −1 (H)).
When L −1 (H) ∼ = K 1,3 , as d(u) = 2, we have ℓ 1 (L m−1 (G)) ≥ 1. By Lemma 2.1, When L −1 (H) ∼ = J 1 or J 2 , as there is no parallel edges in line graphs, m = 1. If L −1 (H) ∼ = J 2 , then G ∼ = J 2 as d(u) = 2, contradicting the definition of G. Then, L −1 (H) ∼ = J 1 . If u = xy is one of the parallel edges of J 1 , then one of end vertices of u, say x, of degree 3 in G satisfies |E G (x) ∩ F | = 2, which implies m ≥ 3 by (6). It is a contradiction with m = 1. When Since d(x) = d(y) = 2 as well as line graphs are claw-free and contain no parallel edges, it shows that m = 2. As d(x) = d(y) = 2, {x, y} ⊆ F and there is a common end vertex of edges x and y of degree three, which shows m ≥ 3 by (6). It contradicts the fact we got before that m = 2.

The k-Triangular Index
Before establishing the bounds for t k (G), we need some lemmas. By the definition of line graphs, if G is a regular graph, then for each integer i ≥ 0, we always have δ(L i (G)) = 2 i (δ(G) − 2) + 2, and so the lower bound in Theorem 3.3 is best possible in this sense.
Lemma 3.4. Let G ∈ G be a simple graph with δ = δ(G). Each of the following holds for each integer i > 0.
Theorem 3.5. Let k ≥ 2 be an integer and G ∈ G be a simple graph with δ = δ(G) and d = d(G). Each of the following holds. (i) Being k-triangular is line graph stable.
Moreover, the equality holds for sufficiently large k when δ ≤ k + 1.
Proof: (i) Suppose G ∈ G is a simple k-triangular graph for given k ≥ 2. Then δ(G) ≥ k + 1 ≥ 3. Pick an edge e 1 e 2 ∈ E(L(G)). To show that L(G) ∈ T k , it is enough to prove that e 1 e 2 lies in at least k distinct triangles in L(G). Let x be the common vertex of e 1 and e 2 in G, and X be the set of all edges adjacent with both edges e 1 and e 2 , that is, . If d(x) ≥ k + 2, then |X| ≥ k. It means that e 1 e 2 lies in at least k distinct triangles in L(G). Now, we consider that d(x) = k + 1. Since G ∈ T k is a simple graph, G[N G (x)] is a complete graph and then e 1 e 2 lies in at least k distinct triangles in L(G).
(ii) Let t = t k (G). First, we consider the situation when δ ≤ 2. As k ≥ 2, by the definition of d, we have t ≥ d. If t < d + 2, then t < d + 1 + ⌈lg k⌉ as k ≥ 2. Assume next that k is so large that t ≥ d + 2.

Proof of Theorem 1.3
An elementary subdivision of a graph G at an edge e = uv is a graph G(e) obtained from G − e by adding a new vertex v e and two new edges uv e and v e v. For a subset X ⊆ E(G), we define G(X) to be the graph obtained from G by elementarily subdividing every edge of X.
Lemma 4.1. For an integer k > 1, if G ∈ G is a k-triangular simple graph and X ⊂ E(G) with |X| = s where 1 ≤ s < k, then G − X ∈ T k−s .
Proof: Pick e ∈ E(G − X). Since G ∈ T k , edge e lies in at least k distinct triangles in G, say C e 1 , C e 2 , . . . , C e k . As E(C e i ∩ C e j ) = {e} for each {i, j} ⊆ [1, k] and |X| = s < k, there exist k − s such triangles C e i ′ where i ′ ∈ [1, k] such that E(C e i ′ ) ∩ X = ∅. It follows that G − X ∈ T k−s . Lemma 4.2. Given two non-negative integers s and t. If G ∈ G is a (s + t + 1)-triangular simple graph, then G is (s, t)-supereulerian.
Then subgraph J contains all edges incident with some v xi , which means that G − Y has a spanning eulerian subgraph J ′ containing X, and so G is (s, t)-supereulerian.
Proof of Theorem 1.3: Combine Theorem 3.1(ii), Theorem 3.5(ii) and Lemma 4.2, and then we complete the proof of it.