Regular subgraphs of dense graphs

Every graph onn vertices, with at leastc _k n logn edges contains ak-regular subgraph. This answers a question of Erdős and Sauer.     

(2,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. with an r-regular spanning subgraph). We show that every τ-tough graph of order n with τ≥2 is (2,k)-factor-critical for every non-negative integer k≤min{2τ−2, n−3}, thus proving a conjecture as well as generalizing the main result of Liu and Yu in [4]. Received: December 16, 1996 / Revised: September 17, 1997     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

(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     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

<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.     

Finding Large p-Colored Diameter Two Subgraphs

Given a coloring of the edges of the complete graph K on n vertices in k colors, a p-colored subgraph of K_n is any subgraph whose edges only use colors from some p element set. We show for k̿ and k\2hphk that there is always a p-colored diameter two subgraph of K_n containing at least [((k+p)n)/(2k)]\displaystyle{(k+p)n \over 2k} vertices and that this is best possible up to an additive constant l satisfying 0hl<k\2.     

DEGREE CONDITIONS OF INDUCED MATCHING EXTENDABLE GRAPHS

Abstract. A simple graph G is induced matching extendable,shortly IM-extendable,if every in-duced matching of G is included in a perfect matching of G. The degree conditions of IM-extend-able graphs are researched in this paper. The main results are as follows：     

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2^{(1/2+∈(k))n} independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory. Research supported in part by the United States-Israel Binational Science Foundation and by a Bergmann Memorial Grant.     

On k-leaf connectivity of a random graph

We prove that, in a random graph with n vertices and N = cn log n edges, the subgraph generated by a set of all vertices of degree at least k + 1 is k-leaf connected for c > 1/4. A threshold function for k-leaf connectivity is also found.     

Large holes in sparse random graphs

We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G _{n, p} ,p=c/n almost surely has a hole of size at least β(c)n asn→∞.     

On extremal sizes of locally k-tree graphs

A graph G is a locally k-tree graph if for any vertex v the subgraph induced by the neighbours of v is a k-tree, k ⩾ 0, where 0-tree is an edgeless graph, 1-tree is a tree. We characterize the minimum-size locally k-trees with n vertices. The minimum-size connected locally k-trees are simply (k + 1)-trees. For k ⩾ 1, we construct locally k-trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an n-vertex locally k-tree graph is between Ω(n) and O(n ²), where both bounds are asymptotically tight. In contrast, the number of edges in an n-vertex k-tree is always linear in n.     

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.     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

Sparse universal graphs for bounded‐degree graphs

Let ℋ︁ be a family of graphs. A graph T is ℋ︁‐universal if it contains a copy of each H ∈ℋ︁ as a subgraph. Let ℋ︁(k,n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an ℋ︁(k,n)‐universal graph T with edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n^2‐2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The property of having a k ‐regular subgraph has a sharp threshold

In this paper it is proved that in the random graph model G(n,p), the property of containing a k ‐regular subgraph, has a sharp threshold for k ≥ 3. It is also shown how to use similar methods to prove, quite easily, the (known fact of) sharpness of having a non empty k ‐core for k ≥ 3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 509–519, 2013     

Regular subgraphs of almost regular graphs

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k ≥ 2q ? 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). It is also proved that every simple graph with maximal degree Δ ≥ 2q ? 2 and average degree $\text{d > (}\text{(2q ? 2)}\text{(2q ? 1)}\text{)(Δ + 1)}$, where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). These results follow from Chevalley's and Olson's theorems on congruences.     

Restricted 2-factor polytopes

The optimal k-restricted 2-factor problem consists of finding, in a complete undirected graph K _n , a minimum cost 2-factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the well-known symmetric travelling salesman problem, and is equivalent to it when ≤k≤n−1. We study the k-restricted 2-factor polytope. We present a large class of valid inequalities, called bipartition inequalities, and describe some of their properties; some of these results are new even for the travelling salesman polytope. For the case k=3, the triangle-free 2-factor polytope, we derive a necessary and sufficient condition for such inequalities to be facet inducing. Received March 4, 1997 / Revised version received September 7, 1998?Published online November 9, 1999     

Packing in regular graphs

A set S of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart in G. The packing number of G, denoted by ρ(G), is the maximum cardinality of a packing in G. Favaron [Discrete Math. 158 (1996), 287–293] showed that if G is a connected cubic graph of order n different from the Petersen graph, then ρ(G) ≥ n/8. In this paper, we generalize Favaron’s result. We show that for k ≥ 3, if G is a connected k-regular graph of order n that is not a diameter-2 Moore graph, then ρ(G) ≥ n/(k² ? 1).     

The maximum valency of regular graphs with given order and odd girth

The odd girth of a graph G is the length of a shortest odd cycle in G. Let d(n, g) denote the largest k such that there exists a k-regular graph of order n and odd girth g. It is shown that dn, g ≥ 2|n/g≥ if n ≥ 2g. As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2j-regular graph of order n and odd girth g if and only if n ≥ gj, where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.