One sufficient condition for Hamiltonian graphs involving distances

In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian.     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

Vertex pancyclicity and new sufficient conditions

For a graph G, ??(G) denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x)?+?d(y)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K _n/2,n/2. In 2001, Xu proved that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+???(G)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K _n/2,n/2. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+?d(w)????n for any three vertices x,y,w of d(x,y)?=?2 and wx or $wy\not\in E(G)$ in G, then G is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results.     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

A note on vertex pancyclic oriented graphs

Let D be an oriented graph of order n ≥ 9, minimum degree at least n − 2, such that, for the choice of distinct vertices x and y, either xy ∈ E(D) or d⁺(x) + d⁻(y) ≥ n − 3. Song (J. Graph Theory 18 (1994), 461–468) proved that D is pancyclic. In this note, we give a short proof, based on Song's result, that D is, in fact, vertex pancyclic. This also generalizes a result of Jackson (J. Graph Theory 5 (1981), 147–157) for the existence of a hamiltonian cycle in oriented graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 313–318, 1999     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

Minimally Non-Preperfect Graphs of Small Maximum Degree

 A graph G is called preperfect if each induced subgraph G ^′⊆G of order at least 2 has two vertices x, y such that either all maximum cliques of G ^′ containing x contain y, or all maximum independent sets of G ^′ containing y contain x, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199–208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d≥3, and contains no 3-edge-connected d ^′-regular subgraph for any 3≤d ^′<d. Received: March 4, 1998 Final version received: August 14, 1999     

Degree sum and nowhere-zero 3-flows

Let G be a simple graph on n vertices. In this paper, we prove that if G satisfies the condition that d(x)+d(y)≥n for each xy∈E(G), then G has no nowhere-zero 3-flow if and only if G is either one of the five graphs on at most 6 vertices or one of a very special class of graphs on at least 6 vertices.     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

Subpancyclicity of line graphs and degree sums along paths

A graph is called subpancyclic if it contains a cycle of length ? for each ? between 3 and the circumference of the graph. We show that if G is a connected graph on n?146 vertices such that d(u)+d(v)+d(x)+d(y)>(n+10/2) for all four vertices u,v,x,y of any path P=uvxy in G, then the line graph L(G) is subpancyclic, unless G is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that L(G) is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

A Fan Type Condition For Heavy Cycles in Weighted Graphs

 A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree d ^w (v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. max{d ^w (x),d ^w (y)∣d(x,y)=2}≥c/2; 2. w(x z)=w(y z) for every vertex z∈N(x)∩N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least c. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result. Received: February 7, 2000 Final version received: June 5, 2001     

Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ? for all 3≤?≤n. In 1972, Erd?s proved that if G is a Hamiltonian graph on n>4k⁴ vertices with independence number k, then G is pancyclic. He then suggested that n=Ω(k²) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n>ck^7/3 suffices.     

On the Structure of Graphs with Bounded Asteroidal Number

 A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d _T (x,y)−d _G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G. Received: July 3, 1998 Final version received: August 10, 1999     

Sur Les Partitions en Chaînes du Graphe Divisoriel

On path partitions of the divisor graph. Let D(x) be the graph with vertices {1, 2, ..., ⌊x⌋} whose edges come from the division relation, and let D(x, y) be the subgraph restricted to the integers with prime factors less than or equal to y. We give sufficient conditions on x and y for the graph D(x, y) to be Hamiltonian. We deduce an asymptotic formula for the number of paths in D(x) needed to partition the set of vertices {1, 2, ..., ⌊x⌋}.

     

A new characterization of L 2(q) by its noncommuting graph

Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L ₂(q) (q = p ⁿ , p is a prime), then G ≅ M.      

New Ore��s Type Results on Hamiltonicity and Existence of Paths of Given Length in Graphs

The well-known Ore??s theorem (see Ore in Am Math Mon 65:55, 1960), states that a graph G of order n such that d(x)?+?d(y)??? n for every pair {x, y} of non-adjacent vertices of G is Hamiltonian. In this paper, we considerably improve this theorem by proving that in a graph G of order n and of minimum degree ????? 2, if there exist at least n ? ?? vertices x of G so that the number of the vertices y of G non-adjacent to x and satisfying d(x)?+?d(y)??? n ? 1 is at most ?? ? 1, then G is Hamiltonian. We will see that there are graphs which violate the condition of the so called ??Extended Ore??s theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007) as well as the condition of Chvatál??s theorem (see Chvátal in J Combin Theory Ser B 12:163?C168, 1972) and the condition of the so called ??Extended Fan?? theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007), but satisfy the condition of our result, which then, only allows to conclude that such graphs are Hamiltonian. This will show the pertinence of our result. We give also a new result of the same type, ensuring the existence of a path of given length.     

Survey on path and cycle embedding in some networks

To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.