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.     

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 degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

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     

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     

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.     

Some extremal results in cochromatic and dichromatic theory

For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {G_n} is a family of graphs where G_n has o(n² log²(n)) edges, then z(G_n) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m² ln²(m)).     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

Distance matrix polynomials of trees

Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d_G on G by letting d_G(x, y) be the minimal length of any path between x and y in G (with d_G(x, x) = 0). Thus, for example, d_G(x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows and columns indexed by the vertex set of G which has d_G(x, y) as its (x, y) entry. In this paper we are concerned with properties of D(G) for the case in which G is a tree (i.e., G is acyclic). In particular, we precisely determine the coefficients of the characteristic polynomial of D(G). This determination is made by deriving surprisingly simple expressions for these coefficients as certain fixed linear combinations of the numbers of various subgraphs of G.     

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.     

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.     

Circuits including a given set of vertices

Let G = (V, E) be a digraph of order n, satisfying Woodall's condition ? x, y ∈ V, if (x, y) ? E, then d⁺(x) + d^?(y) ≥ n. Let S be a subset of V of cardinality s. Then there exists a circuit including S and of length at most Min(n, 2s). In the case of oriented graphs we obtain the same result under the weaker condition d⁺(x) + d^?(y) ≥ n – 2 (which implies hamiltonism).     

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and p_ij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff $\text{dim}\text{(x ? y) = s ? 1}$. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p₁₁(x, y) = λ whenever d(x, y) = 1, and p₁₁(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where $\text{2d}\text{3}\text{≤ s < d ? 2}$ and $\text{(s, q, d) ≠ (}\text{2d}\text{3}\text{, 2, d)}$, are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.     

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     

Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G.     

Covering a graph with cycles passing through given edges

We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e₁, …, e_k of G, there exist k vertex-disjoint cycles C₁, …, C_k in G such that e_i ∈ E(C_i) for all i ∈ {1, …, k} and V(C₁ ∪ ···∪ C_k) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997     

Group Connectivity of Bridged Graphs

Let G be a 2-edge-connected simple graph, and let A denote an abelian group with the identity element 0. If a graph G ^* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to G*. A graph G is bridged if every cycle of length at least 4 has two vertices x, y such that d _G (x, y) < d _C (x, y). In this paper, we investigate the group connectivity number Λ _g (G) = min{n: G is A-connected for every abelian group with |A| ≥ n} for bridged graphs. Our results extend the early theorems for chordal graphs by Lai (Graphs Comb 16:165–176, 2000) and Chen et al. (Ars Comb 88:217–227, 2008).     

On 2-Detour Subgraphs of the Hypercube

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d _H (x, y) ≤ d _G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q _n has at least clog₂ n · 2 ⁿ edges. József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398. Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.     

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.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.