Hamiltonian Line Graphs Research Articles

Complete family reduction and spanning connectivity in line graphs

Complete families of connected graphs, introduced by Catlin in the 1980s, have been known useful in the study of certain graphical properties that are closed under taking contractions. We show that given any complete family C of connected graphs such that C contains graphs with sufficiently many edge-disjoint spanning trees, for any real number a and b with 0<a<1, there exists a finite obstacle family F=F(a,b,C) such that for any simple graph G on n vertices satisfying the Ore-type degree conditionmin⁡{dG(u)+dG(v):u,v∈V(G) and uv∉E(G)}≥an+b, either G∈C or G can be contracted to a member in F. This result is applied to the study of spanning connectivity of line graphs. The spanning connectivity is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with u≠v, G has a spanning subgraph H consisting of k internally disjoint (u,v)-paths. Z. Ryjáček and P. Vrána in [J. Graph Theory, 66 (2011) 152-173] prove that a fascinating conjecture of Thomassen on hamiltonian line graphs is equivalent to that every essentially 4-edge-connected graph has a 2-spanning-connected line graph. We prove that for any essentially 3-edge-connected graph G and any positive integer s, if G satisfies an Ore degree condition lower bounded by an arbitrary linear function in the number of vertices, then L(G) is s-spanning-connected with only finitely many contraction obstacles. When s=3, we determine a finite graph family J′(n) such that for every simple graph G on n≥156 vertices with κ(L(G))≥3 and satisfying d(u)+d(v)≥2(n−6)5 for any pair of nonadjacent vertices u and v, we have either κ⁎(L(G))≥3 or G is contractible to a member in J′(n).

Hamiltonian line graphs with local degree conditions

Let N1,1,1 be the graph formed by attaching a pendant edge to each vertex of a triangle, and B1,2 be a graph obtained by attaching end vertices of two disjoint paths of lengths 1,2 to two vertices of a triangle. Broersma (1993) [2] and Čada et al. (2016) [3] conjectured that for a 2-connected claw-free simple graph G and for a fixed graph Γ∈{N1,1,1,B1,2,P6}, if δΓ(G)=min⁡{dG(v):dH(v)=1 for any induced subgraph H≅Γ in G}≥|V(G)|−23, then G is Hamiltonian. While Chen settles this conjecture recently, the following two results of the conjecture for 3-connected line graphs are proved.(i) For real numbers a,b with 0<a<1, there exists a family F(a,b) of finitely many nonsupereulerian graphs, such that for any 3-connected line graph H=L(G) of a simple graph G, if δN1,1,1(H)≥a|V(H)|+b, then either H is Hamiltonian or G is contractible to a member in F(a,b).(ii) Let H=L(G) be a 3-connected line graph of a simple graph G with |V(H)|≥116. If δN1,1,1(H)≥|V(H)|+510, then either H is Hamiltonian or G is isomorphic to the graph P(10)′, which is formed from the Petersen graph P(10) by attaching |V(H)|−1510 pendant edges to every vertex of P(10).

On hamiltonian line graphs of hypergraphs

AbstractA graph is supereulerian if it has a spanning eulerian subgraph. Harary and Nash‐Williams in 1968 proved that the line graph of a graph is hamiltonian if and only if has a dominating eulerian subgraph, Jaeger in 1979 showed that every 4‐edge‐connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge‐disjoint spanning trees is a contractible configuration for supereulerianicity. Utilizing the notion of partition‐connectedness of hypergraphs introduced by Frank, Király, and Kriesell in 2003, we generalize the above‐mentioned results of Harary and Nash‐Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are hamiltonian, and showing that every 2‐partition‐connected hypergraph is a contractible configuration for supereulerianicity.

Catlin’s reduced graphs with small orders

A graph is supereulerian if it has a spanning closed trail. Catlin in 1990 raised the problem of determining the reduced nonsupereulerian graphs with small orders, as such results are of particular importance in the study of Eulerian subgraphs and Hamiltonian line graphs. We determine all reduced graphs with order at most 14 and with few vertices of degree 2, extending former results of Chen and Chen in 2016. In 1985, Bauer proposed the problems of determining best possible sufficient conditions on minimum degree of a simple graph (or a simple bipartite graph, respectively) G to ensure that its line graph L(G) is Hamiltonian. These problems have been settled by Catlin and Lai in 1988, respectively. As an application of our main results, we prove the following for a connected simple graph G on n vertices: If then for sufficiently large n, L(G) is Hamilton-connected if and only if both and G is not nontrivially contractible to the Wagner graph. If G is bipartite and then for sufficiently large n, L(G) is Hamilton-connected if and only if both and G is not nontrivially contractible to the Wagner graph.

Lai's Conditions for Spanning and Dominating Closed Trails

A graph is supereulerian if it has a spanning closed trail. For an integer $r$, let ${\cal Q}_0(r)$ be the family of 3-edge-connected nonsupereulerian graphs of order at most $r$. For a graph $G$, define $\delta_L(G)=\min\{\max\{d(u), d(v) \}| \ \mbox{ for any $uv\in E(G)$} \}$. For a given integer $p\ge 2$ and a given real number $\epsilon$, a graph $G$ of order $n$ is said to satisfy a Lai's condition if $\delta_L(G)\ge \frac{n}{p}-\epsilon$. In this paper, we show that if $G$ is a 3-edge-connected graph of order $n$ with $\delta_L(G)\ge \frac{n}{p}-\epsilon$, then there is an integer $N(p, \epsilon)$ such that when $n&gt; N(p,\epsilon)$, $G$ is supereulerian if and only if $G$ is not a graph obtained from a graph $G_p$ in the finite family ${\cal Q}_0(3p-5)$ by replacing some vertices in $G_p$ with nontrivial graphs. Results on the best possible Lai's conditions for Hamiltonian line graphs of 3-edge-connected graphs or 3-edge-connected supereulerian graphs are given, which are improvements of the results in [J. Graph Theory 42(2003) 308-319] and in [Discrete Mathematics, 310(2010) 2455-2459].

On s‐Hamiltonian Line Graphs

AbstractFor an integer s ≥ 0, a graph G is s‐hamiltonian if for any vertex subset with |S′| ≤ s, G ‐ S′ is hamiltonian. It is well known that if a graph G is s‐hamiltonian, then G must be (s+2)‐connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s‐hamiltonian if and only if it is (s+2)‐connected.

On 3-connected hamiltonian line graphs

Thomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and Vumar (2009) [5] found another family of graphs, called P3D graphs, which properly contain all quasi-claw-free graphs. In this paper, we investigate the hamiltonicity of 3-connected line graphs of DCT graphs and P3D graphs, and prove that if G is a DCT graph or a P3D graph with κ(L(G))≥3 and if L(G) does not have an independent vertex 3-cut, then L(G) is hamiltonian. Consequently, every 4-connected line graph of a DCT graph or a P3D graph is hamiltonian.

Traceability of line graphs

Brualdi and Shanny [R.A. Brualdi, R.F. Shanny, Hamiltonian line graphs, J. Graph Theory 5 (1981) 307–314], Clark [L. Clark, On hamitonian line graphs, J. Graph Theory 8 (1984) 303–307] and Veldman [H.J. Veldman, On dominating and spanning circuits in graphs, Discrete Math. 124 (1994) 229–239] gave minimum degree conditions of a line graph guaranteeing the line graph to be hamiltonian. In this paper, we investigate the similar conditions guaranteeing a line graph to be traceable. In particular, we show the following result: let G be a simple graph of order n and L ( G ) its line graph. If n is sufficiently large and, either δ ( L ( G ) ) > 2 ⌊ n − 8 4 ⌋ ; or δ ( L ( G ) ) > 2 ⌊ n − 20 10 ⌋ and G is almost bridgeless, then L ( G ) is traceable. As a byproduct, we also show that every 2-edge-connected triangle-free simple graph with order at most 9 has a spanning trail. These results are all best possible.

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

AbstractIn this paper, we show that if G is a 3‐edge‐connected graph with $S \subseteq V(G)$ and $|S| \le 12$, then either G has an Eulerian subgraph H such that $S \subseteq V(H)$, or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any $|S|\le 23$, G has an Eulerian subgraph H such that $S\subseteq V(H)$. As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003

Line Graphs and Forbidden Induced Subgraphs

Beineke and Robertson independently characterized line graphs in terms of nine forbidden induced subgraphs. In 1994, Šoltés gave another characterization, which reduces the number of forbidden induced subgraphs to seven, with only five exceptional cases. A graph is said to be a dumbbell if it consists of two complete graphs sharing exactly one common edge. In this paper, we show that a graph with minimum degree at least seven that is not a dumbbell is a line graph if and only if it does not contain three forbidden induced subgraphs including K1, 3 and K5−e. Applications of our main results to other forbidden induced subgraph characterizations of line graphs and to hamiltonian line graphs are also discussed.

On subpancyclic line graphs

We give a best possible Dirac-like condition for a graph G so that its line graph L(G) is subpancyclic, i.e., L(G) contains a cycle of length l for each l between 3 and the circumference of G. The result verifies the conjecture posed by Xiong (Pancyclic results in hamiltonian line graphs, in: Combinatorics and Graph Theory '95, vol. 2, Proceedings of the Summer School and International Conference on Combinatorics, World Scientific). © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 67–74, 1998

Eulerian subgraphs containing given vertices and hamiltonian line graphs

Let G be a graph and let D 1( G) be the set of vertices of degree 1 in G. Veldman (1994) proves the following conjecture from Benhocine et al. (1986) that if G − D 1( G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E( G), d( x) + d( y) > (2 n)/5 − 2, then for n large, L( G), the line graph of G, is hamiltonian. We shall show the following improvement of this theorem: if G − D 1( G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E( G), max[; d( x), d( y)] ⩾ n/5 − 1, then for n large, L( G) is hamiltonian with the exception of a class of well characterized graphs. Our result implies Veldman's theorem.

Hamiltonian Properties of Grid Graphs

This paper presents sufficient conditions for a grid graph to be Hamiltonian. It is proved that all finite grid graphs of positive width have Hamiltonian line graphs.

Supereulerian graphs: A survey

AbstractA graph is supereulerian if it has a spanning eulerian subgraph. There is a rduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method, and its applications.

On hamiltonian line graphs and connectivity

A well-known conjecture of Thomassen says that every 4-connected line graph is hamiltonian. In this paper we prove that every 7-connected line graph is hamiltonian-connected.

Reduced graphs of diameter two

AbstractA graph H is collapsible if for every subset X ⊆ V(H), H has a spanning connected subgraph whose set of odd‐degree vertices is X. In any graph G there is a unique collection of maximal collapsible subgraphs, and when all of them are contracted, the resulting contraction of G is a reduced graph. Interest in reduced graphs arises from the fact [4] that a graph G has a spanning closed trail if and only if its corresponding reduced graph has a spanning closed trail. The concept can also be applied to study hamiltonian line graphs [11] or double cycle covers [8]. In this article, we characterize the reduced graphs of diameter two. As applications, we obtain prior results in [12] and [14], and show that every 2‐edge‐connected graph with diameter at most two either admits a double cycle cover with three even subgraphs or is isomorphic to the Petersen graph.

A result on Hamiltonian line graphs involving restrictions on induced subgraphs

AbstractIt is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph.

Contractions and hamiltonian line graphs

AbstractUsing the contraction method, we find a best possible condition involving the minimum degree for a triangle‐free graph to have a spanning eulerian subgraph.

Hamiltonian line graphs

AbstractSufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle.

On hamiltonian line-graphs

Introduction. The line-graph L(G) of a nonempty graph G is the graph whose point set can be put in one-to-one correspondence with the line set of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of G are adjacent. In this paper graphs whose line-graphs are eulerian or hamiltonian are investigated and characterizations of these graphs are given. Furthermore, necessary and sufficient conditions are presented for iterated line-graphs to be eulerian or hamiltonian. It is shown that for any connected graph G which is not a path, there exists an iterated line-graph of G which is hamiltonian.