The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

For a connected graph G = (V, E), an edge set S ì E{S\subset E} is called a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λ _k (G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets U₁,U₂ ì V(G){U_1,U_2\subset V(G)}, denote the set of edges of G with one end in U ₁ and the other in U ₂ by [U ₁, U ₂]. Define x_k(G)=min{|[U,V(G)\ U]|: U{\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U} is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λ _k -optimal if λ _k (G) = ξ _k (G). A graph is said to be super-λ _k if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λ _k -optimal or super-λ _k . Moreover, we demonstrate that our results are best possible.     

Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V(G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is super-λ if and only if G\ncong K₂{G\ncong K_2}.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

Super-connectivity and super-edge-connectivity for some interconnection networks

Let G=(V,E) be a k-regular graph with connectivity κ and edge connectivity λ. G is maximum connected if κ=k, and G is maximum edge connected if λ=k. Moreover, G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is {x|(v,x)E} for some vertex vV; and G is super-edge-connected if it is maximum edge connected and every minimum edge disconnecting set is {(v,x)|(v,x)E} for some vertex vV. In this paper, we present three schemes for constructing graphs that are super-connected and super-edge-connected. Applying these construction schemes, we can easily discuss the super-connected property and the super-edge-connected property of hypercubes, twisted cubes, crossed cubes, möbius cubes, split-stars, and recursive circulant graphs.     

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.     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Contractible Triples in Highly Connected Graphs

We prove that if G is k-connected (with k ≥ 2), then G contains either a cycle of length 4 or a connected subgraph of order 3 whose contraction results in a k-connected graph. This immediately implies that any k-connected graph has either a cycle of length 4 or a connected subgraph of order 3 whose deletion results in a (k − 1)-connected graph.     

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.     

On the 3-restricted edge connectivity of permutation graphs

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G−W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge connectivity of a permutation graph is upper bounded by the so-called minimum k-edge degree. In this paper some sufficient conditions guaranteeing optimal k-restricted edge connectivity and super k-restricted edge connectivity for permutation graphs are presented for k=2,3.     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Some results on R 2-edge-connectivity of even regular graphs

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an Rredge-cut if G-S is disconnected and comains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ^n(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ^n(G)=g(2k-2) for any 2k-regular connected graph G (≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

Binding Numbers and Connected Factors

In this paper, we obtain some sufficient conditions based on binding number for a graph to have a connected factor with degree restrictions. Let α and k be positive integers with α + k ≥ 4. Let G be a connected graph with a spanning subgraph F, each component of which has order at least α. We show that if the binding number of G is greater than (α k − α)/(α k − α −1) (k ≥ 2) and α/(α −2) (k = 1) then G has a connected subgraph which has F and in which every vertex v has degree at most deg _F (v) + k. From the result, we derive various sufficient conditions for a graph to have a connected [a, b]-factor.     

Goal-minimally <Emphasis Type="Italic">k</Emphasis>-elongated graphs

Let k be an integer. A 2-edge connected graph G is said to be goal-minimally k-elongated (k-GME) if for every edge uv ∈ E(G) the inequality d _G−uv (x, y) > k holds if and only if {u, v} = {x, y}. In particular, if the integer k is equal to the diameter of graph G, we get the goal-minimally k-diametric (k-GMD) graphs. In this paper we construct some infinite families of GME graphs and explore k-GME and k-GMD properties of cages. This research was supported by the Slovak Scientific Grant Agency VEGA No. 1/0406/09.     

Bound on <Emphasis Type="Italic">m</Emphasis>-restricted Edge Connectivity

An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restrict edge connectivity λm is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let θ(X) denote the number of edges with one end in X and the other not in X and ξm=min{θ(X) ;X is a connected vertex-induced subgraph of order m}.It is proved in this paper that if G has girth at least m/2 2,then λm≤ξm.The upper bound of λm is sharp.     

3‐Flows and Combs

Tutte's 3‐Flow Conjecture states that every 2‐edge‐connected graph with no 3‐cuts admits a 3‐flow. The 3‐Flow Conjecture is equivalent to the following: let G be a 2‐edge‐connected graph, let S be a set of at most three vertices of G; if every 3‐cut of G separates S then G has a 3‐flow. We show that minimum counterexamples to the latter statement are 3‐connected, cyclically 4‐connected, and cyclically 7‐edge‐connected.     

Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

An <Emphasis Type="Italic">s</Emphasis>-Hamiltonian Line Graph Problem

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.     

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     

(3,k)-Factor-Critical Graphs and Toughness

 A graph is (r,k)-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. an r-regular spanning subgraph). Let t(G) denote the toughness of graph G. In this paper, we show that if t(G)≥4, then G is (3,k)-factor-critical for every non-negative integer k such that n+k even, k<2 t(G)−2 and k≤n−7. Revised: September 21, 1998