Maximum Number Of Edges Research Articles

On Graphs with a Unique Perfect Matching

If a graph has a unique perfect matching, we call it a UPM-graph. In this paper we study UPM-graphs. It was shown by Kotzig that a connected UPM-graph has a cut edge belonging to its unique perfect matching. We strengthen this result to a further structural characterization. Using the stronger result, we present a characterization of claw-free UPM-graphs, and prove that for any fixed positive integer $$n$$n, the number of edges of saturated UPM-graphs on $$2n$$2n vertices form an arithmetic progression from $$(2n+2)\lfloor \log _2(n+1)\rfloor -2^{2+\lfloor \log _2(n+1)\rfloor }+n+4$$(2n+2)?log2(n+1)?-22+?log2(n+1)?+n+4 to $$n^2$$n2 with common difference 2. For a fixed positive integer $$n$$n, we determine the number of labelled UPM-trees on $$2n$$2n vertices. For a bipartite UPM-graph which has maximum number of edges, we determine the number of spanning UPM-trees of it.

Fast algorithms for some dominating induced matching problems

We describe O(n) time algorithms for finding the minimum weighted dominating induced matching of chordal, dually chordal, biconvex, and claw-free graphs. For the first three classes, we prove tight O(n) bounds on the maximum number of edges that a graph having a dominating induced matching may contain. By applying these bounds, and employing existing O(n+m) time algorithms we show that they can be reduced to O(n) time. For claw-free graphs, we describe a variation of the existing algorithm for solving the unweighted version of the problem, which decreases its complexity from O(n2) to O(n), while additionally solving the weighted version. The same algorithm can be easily modified to count the number of DIM's of the given graph.

Random Linear Network Coding with Probabilistic Polynomial-Time Wiretap Adversary

Linear network coding can achieve the network capacity in the single source multicast case. For the single source multicast network with wiretap adversary, the paper of Cai and Yeung constructs an information-secure linear coding which tightly achieves the transmitting rate upper bound , where is the capacity of the network and is the maximum number of edges the adversary can wiretap. The current paper considers the cryptographic security for the messages in the random linear network coding setting. The indistinguishability under chosen-message-attack (or CMA-Security) from probabilistic polynomial-time (PPT) wiretap adversary is defined. With a pseudorandom generator against linear cryptanalysis, an end-to-end protocol is constructed for multicast network using random linear network coding. When the adversary wiretap less than edges, the protocol is CMA-Secure and attains transmitting rate approaching the network’s capacity from below, asymptotically in the expanding ratio (i.e., length of output string/length of input string) of the pseudorandom generator. Compared with secret-key based scheme, no secret is needed by the receiver nodes, which means the key managing issues for dynamic networks would never come up.

Induced Decompositions of Highly Dense Graphs

Given two graphs F and G, an inducedF-decomposition of G is a partition of EG into induced subgraphs isomorphic to F. Bondy and Szwarcfiter [J. Graph Theory, DOI: 10.1002/jgt.21654] defined the value ex*n,F as the maximum number of edges in a graph of order n admitting an inducedF-decomposition and determined the value of ex*n,F for some graphs and families of graphs. In this article, we prove that ex*n,F=n2-on2 is valid for all graphs F. We also present tighter asymptotic bounds for some of the small graphs for which the exact value of ex* remains unknown. The proofs are based on the heavy use of various classes of Kneser graphs and hypergraphs.

Equicovering Subgraphs of Graphs and Hypergraphs

As a variation on the $t$-Equal Union Property ($t$-EUP) introduced by Lindström, we introduce the $t$-Equal Valence Property ($t$-EVP) for hypergraphs: a hypergraph satisfies the $t$-EVP if there are $t$ pairwise edge-disjoint subhypergraphs such that for each vertex $v$, the degree of $v$ in all $t$ subhypergraphs is the same. In the $t$-EUP, the subhypergraphs just have the same sets of vertices with positive degree. For both the $2$-EUP and the $2$-EVP, we characterize the graphs satisfying the property and determine the maximum number of edges in a graph not satisfying it. We also study the maximum number of edges in both $k$-uniform and general hypergraphs not satisfying the $t$-EVP.

Heuristics for the Maximum 2-Layer RAC Subgraph Problem

A 2-layer drawing of a bipartite graph G is a drawing such that the vertices of each partition set are drawn as points of a distinct horizontal line (called a layer) and the edges are drawn as straight-line segments. We study 2-layer drawings where edges can cross only at right angles; these drawings are called 2-layer right angle crossing drawings (2-layer RAC drawings for short). We focus on the following problem, which we call the maximum 2-layer RAC subgraph (M2LRacS) problem. Given a bipartite graph G, compute a subgraph H of G such that: (i) H admits a 2-layer RAC drawing and (ii) H has the maximum number of edges among the subgraphs of G that satisfy (i). We study this problem both in the no-fixed-layer setting, where no restriction is given on the vertex ordering on each layer, and in the 1-fixed-layer setting, where the ordering of the vertices of one of the two layers is given as part of the input and cannot be changed. The M2LRacS problem is known to be 𝒩𝒫-hard in the no-fixed-layer setting (Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2011) 2-Layer Right Angle Crossing Drawings. Proc. IWOCA 2011, Lecturer Notes in Computer Science 7056, pp. 156–169; Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2014) 2-layer right angle crossing drawings. Algorithmica, 68, 954–997), but no algorithm has been proposed so far to solve it. We prove that the M2LRacS problem remains 𝒩𝒫-hard even in the 1-fixed-layer setting, and provide different heuristics to solve it in the two settings; one of these heuristics is a 3-approximation algorithm for the no-fixed-layer setting. Also, we present the results of an experimental study that compares our heuristics and shows the effectiveness of the 3-approximation algorithm in practice.

P- vertex dominating graph and p- vertex domination number

A p-vertex dominating graph is a finite, simple and connected graph G with exactly p-vertices adjacent to all the vertices of G. We characterize 1-vertex dominating graph. We construct 1-vertex dominating graphs with maximum number of edges and prove that they are hamiltonian, 2-connected, triple connected and 1-hamiltonian. More over we define a 1-vertex dominating set D1v and the 1-vertex domination number γ1v (G) of a graph G. Also we find the 1-vertex domination number for some graphs.

Turán numbers of bipartite graphs plus an odd cycle

For an odd integer k, let Ck={C3,C5,…,Ck} denote the family of all odd cycles of length at most k and let C denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family F of bipartite graphs, there exists k such that ex(n,F∪Ck)∼ex(n,F∪C) as n→∞. This conjecture was proved by Erdős and Simonovits when F={C4}, and for certain families of even cycles in [14]. In this paper, we give a general approach to the conjecture using Scott's sparse regularity lemma. Our approach proves the conjecture for complete bipartite graphs K2,t and K3,3: we obtain more strongly that for any odd k⩾5,ex(n,F∪{Ck})∼ex(n,F∪C) and we show further that the extremal graphs can be made bipartite by deleting very few edges. In contrast, this formula does not extend to triangles – the case k=3 – and we give an algebraic construction for odd t⩾3 of K2,t-free C3-free graphs with substantially more edges than an extremal K2,t-free bipartite graph on n vertices. Our general approach to the Erdős–Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n bipartite F-free graph.

Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length l, denoted by ℙ l (k) , is a family of k-sets {F 1,...,F l such that |F i ∩ F i+1|=1 for each i and F i ∩ F bj = $\not 0$ whenever |i−j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turan number of H, denoted by ex k (n, H), is the maximum number of edges in a k-uniform hypergraph $\mathcal{F}$ on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex k (n, P l (k) exactly for all fixed l ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.

Maximum hypergraphs without regular subgraphs

We show that an n-vertex hypergraph with no r-regular subgraphs has at most 2n−1 + r− 2 edges. We conjecture that if n > r, then every n-vertex hypergraph with no r-regular subgraphs having the maximum number of edges contains a full star, that is, 2n−1 distinct edges containing a given vertex. We prove this conjecture for n ≥ 425. The condition that n > r cannot be weakened.

Turán Numbers for Forests of Paths in Hypergraphs

The Turán number of an $r$-uniform hypergraph $H$ is the maximum number of edges in any $r$-graph on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{P}_{\ell}^{(r)}$ denote the family of $r$-uniform loose paths on $\ell$ edges, $\mathcal{F}(k,l)$ denote the family of hypergraphs consisting of $k$ disjoint paths from $\mathcal{P}_{\ell}^{(r)}$, and $L_\ell^{(r)}$ denote an $r$-uniform linear path on $\ell$ edges. We determine precisely ${ex}_r(n;\mathcal{F}(k,l))$ and ${ex}_r(n;k\cdot L_\ell^{(r)})$, as well as the Turán numbers for forests of paths of differing lengths (whether these paths are loose or linear) when $n$ is appropriately large dependent on $k,l,r$ for $r\geq 3$. Our results build on recent results of Füredi, Jiang, and Seiver, who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turán numbers are exactly determined.

The Turàn number of the graph 3P4

AbstractLet ex (n,G) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let P

Edge-maximality of power graphs of finite cyclic groups

We show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph. Contains corrections to published version.

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C k denote a cycle of length k, and let C k denote the set of cycles C ℓ, where 3≤ℓ≤k and ℓ and k have the same parity. Erdős and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C k ) ∼ z(n,F) — here we write f(n) ∼ g(n) for functions f,g: ℕ → ℝ if lim n→∞ f(n)/g(n)=1. They proved this when F ={C 4} by showing that ex(n,{C 4;C 5})∼z(n,C 4). In this paper, we extend this result by showing that if ℓ∈{2,3,5} and k>2ℓ is odd, then ex(n,C 2ℓ ∪{C k }) ∼ z(n,C 2ℓ ). Furthermore, if k>2ℓ+2 is odd, then for infinitely many n we show that the extremal C 2ℓ ∪{C k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2ℓ, and furthermore the asymptotic result does not hold when (ℓ,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.

Induced matchings in subcubic graphs without short cycles

An induced matching of a graph G is a set M of edges of G such that every two vertices of G that are incident with distinct edges in M have distance at least 2 in G. The maximum number of edges in an induced matching of G is the induced matching number νs(G) of G. In contrast to ordinary matchings, induced matchings in graphs are algorithmically difficult. Next to hardness results and efficient algorithms for restricted graph classes, lower bounds are therefore of interest.We show that if G is a connected graph of order n(G), maximum degree at most 3, girth at least 6, and without a cycle of length 7, then νs(G)≥n(G)−15, and we characterize the graphs achieving equality in this lower bound.

New estimates in the problem of the number of edges in a hypergraph with forbidden intersections

We improve the Frankl-Wilson upper bound on the maximal number of edges in a hypergraph with forbidden cardinalities of edge intersections.

The Maximum Number of Edges in Geometric Graphs with Pairwise Virtually Avoiding Edges

Let G be a geometric graph on n vertices that are not necessarily in general position. Assume that no line passing through one edge of G meets the relative interior of another edge. We show that in this case the number of edges in G is at most 2n?3.

Anti-Ramsey number of matchings in hypergraphs

A k-matching in a hypergraph is a set of k edges such that no two of these edges intersect. The anti-Ramsey number of a k-matching in a complete s-uniform hypergraph H on n vertices, denoted by ar(n,s,k), is the smallest integer c such that in any coloring of the edges of H with exactly c colors, there is a k-matching whose edges have distinct colors. The Turán number, denoted by ex(n,s,k), is the the maximum number of edges in an s-uniform hypergraph on n vertices with no k-matching. For k≥3, we conjecture that if n>sk, then ar(n,s,k)=ex(n,s,k−1)+2. Also, if n=sk, then ar(n,s,k)={ex(n,s,k−1)+2if k<csex(n,s,k−1)+s+1if k≥cs, where cs is a constant dependent on s. We prove this conjecture for k=2,k=3, and sufficiently large n, as well as provide upper and lower bounds.

On the Turán Number of Forests

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.

A Polynomial-Time Algorithm for Computing the Maximum Common Connected Edge Subgraph of Outerplanar Graphs of Bounded Degree

The maximum common connected edge subgraph problem is to find a connected graph with the maximum number of edges that is isomorphic to a subgraph of each of the two input graphs, where it has applications in pattern recognition and chemistry. This paper presents a dynamic programming algorithm for the problem when the two input graphs are outerplanar graphs of a bounded vertex degree, where it is known that the problem is NP-hard, even for outerplanar graphs of an unbounded degree. Although the algorithm repeatedly modifies input graphs, it is shown that the number of relevant subproblems is polynomially bounded, and thus, the algorithm works in polynomial time.