Maximum Number Of Edges Research Articles

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Dense circuit graphs and the planar Turán number of a cycle

AbstractThe planar Turán number of a graph is the maximum number of edges in an ‐vertex planar graph without as a subgraph. Let denote the cycle of length . The planar Turán number is known for . We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant so that for all and . When this bound is tight up to the constant and proves a conjecture of Cranston, Lidický, Liu, and Shantanam.

Regular Subgraphs of Linear Hypergraphs

Abstract We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Łuczak, Mubayi, Nagle, Person, Rödl, Schacht, and Verstraëte. We use this result to show that the maximum number of edges in a $3$-uniform hypergraph on $n$ vertices containing no immersion of a closed surface is $n^{2+o(1)}$. Furthermore, we present results on the maximum number of edges in $k$-uniform linear hypergraphs containing no $r$-regular subhypergraph.

The Complexity of Finding and Enumerating Optimal Subgraphs to Represent Spatial Correlation

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee et al. (Stat Comput 31(4):51, 2021. https://doi.org/10.1007/s11222-021-10025-7) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper, we address the computational complexity of the associated graph optimisation problem. We demonstrate that this optimisation problem is NP-hard; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem. The first is parameterised by both treewidth and maximum degree, while the second is parameterised by the maximum number of edges that can be removed and is also restricted to settings where the input graph has maximum degree three. Both of these algorithms solve not only the decision problem, but also enumerate all solutions with polynomial time precalculation, delay, and postcalculation time in respective restricted settings. For this problem, efficient enumeration allows the uncertainty in the spatial correlation to be utilised in the modelling. The first enumeration algorithm utilises dynamic programming on a tree decomposition of the input graph, and has polynomial time precalculation and linear delay if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but can output all solutions with FPT precalculation time and linear delay when the maximum number of edges that can be removed is taken as the parameter.

Planar Turán Number of the Θ6

Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.

Graphs, Disjoint Matchings and Some Inequalities

A graph G is k-edge-colorable if the edges of G can be assigned a color from {1, ..., k} so that adjacent edges of G receive different colors. A maximum kedge-colorable subgraph of G is a k-edge-colorable subgraph of G containing maximum number of edges. For k ≥ 1 and a graph G, let νk(G) denote the number of edges in a maximum k-edge-colorable subgraph of G. In 2010 Mkrtchyan, Petrosyan and Vardanyan proved that if G is a cubic graph, then ν2(G) ≤ |V (G)|+2·ν3(G) 4 [13]. For cubic graphs containing a perfect matching, in particular, for bridgeless cubic graphs, this inequality can be stated as ν2(G) ≤ ν1(G)+ν3(G) 2 . One may wonder whether there are other well-known graph classes, where a similar result can be obtained. In this work, we prove lower bounds for νk(G) in terms of νk−1(G)+νk+1(G) 2 for k ≥ 2 and graphs G containing at most 1 cycle. We also present the corresponding conjectures for nearly bipartite graphs.

Optimal Layout of (Kp − Cp)n into Wheel-Like Networks

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), https://doi.org/10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

On the number of edges in a K5-minor-free graph of given girth

We consider the following question: what is the maximum number of edges in a K5-minor-free graph with n vertices and girth g, where n≥4? With no restriction on the girth (g=3), it is well known that the answer is 3n−6. For g=4 and n≥5, the answer is known to be 3n−9. For g=5, we show there are two graphs with 9n−205 edges and every other graph contains at most 9n−215 edges and equality holds for infinitely many graphs. For g=8 and large n the answer is 3n−92. For g=4k, k≥2, we conjecture the answer is 3k3k−2(n−3).

Extremal Numbers and Sidorenko’s Conjecture

Abstract Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\textrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

Results on Extremal Graph Theoretic Questions for Q-Ary Vectors

A 𝑞-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 × 𝑛 matrix with entries from {0, … , 𝑞}, such that each row of the matrix (called a 𝑞-edge) contains exactly two nonzero entries. If 𝐻 is a 𝑞-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of 𝑞-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, 𝑞, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex 𝑞-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, 𝑞, 𝑞 + 1) for even 𝑞 to the case 𝑞 = 2, and determine the asymptotics of ex(𝑛, 𝐶2𝑘+1, 𝑞, 𝑞 + 1).

Wickets in 3-uniform hypergraphs

In this note, we consider a Turán-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let Hn(3) be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called wicket, is formed by three rows and two columns of a 3×3 point matrix. We describe two linear hypergraphs that if we forbid either of them in Hn(3), then the hypergraph is sparse, i.e. the number of its edges is o(n2). Since both contain a wicket, it implies a conjecture of Gyárfás and Sárközy that wicket-free hypergraphs are sparse.

Graph theoretic properties of good sets in hypercube

A good set on k vertices is a vertex induced subgraph of the hypercube Q n that has the maximum number of edges. In this paper we discuss graph theoretic properties of good sets including connectivity, vertex decomposition, domination and identifying path covers of good sets.

The maximum 2-edge-colorable subgraph problem and its fixed-parameter tractability

A $k$-edge-coloring of a graph is an assignment of colors $\{1,...,k\}$ to edges of the graph such that adjacent edges receive different colors. In the maximum $k$-edge-colorable subgraph problem we are given a graph and an integer $k$, the goal is to find a $k$-edge-colorable subgraph with maximum number of edges together with its $k$-edge-coloring. In this paper, we consider the maximum 2-edge-colorable subgraph problem and present some results that deal with the fixed-parameter tractability of this problem.

Order Structure of Good Sets in Hypercube

A good set on k vertices is a vertex induced subgraph of the hypercube Qn that has the maximum number of edges. The long-lasting problem of characterizing graphs that are cover graphs of lattices is NP-complete. This paper constructs and studies lattice theoretic properties of a class of lattices whose cover graphs are isomorphic to good sets.

Formulations for the maximum common edge subgraph problem

The Maximum Common Edge Subgraph Problem (MCES), given two graphs G and H, asks about finding a maximum common subgraph between those two graphs with a maximal number of edges. We present new integer programming formulations for MCES and compare the performances of those formulations with earlier formulations from the literature.

On the Halin Turán number of short cycles

A Halin graph is a graph constructed by embedding a tree with no vertex of degree two in the plane and then adding a cycle to join the tree’s leaves. The Halin Turán number of a graph F , denoted as exH(n, F ), is the maximum number of edges in an n-vertex Halin graph. In this paper, we give the exact value of exH(n, C4), where C4 is a cycle of length 4. We also pose a conjecture for the Halin Turán number of longer cycles.

Exact values for some unbalanced Zarankiewicz numbers

AbstractFor positive integers , , and , the Zarankiewicz number is defined to be the maximum number of edges in a bipartite graph with parts of sizes and that has no complete bipartite subgraph containing vertices in the part of size and vertices in the part of size . A simple argument shows that, for each , when . Here, for large , we determine the exact value of in almost all of the remaining cases where . We establish a new family of upper bounds on which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs.

Ramsey–Turán problems with small independence numbers

Given a graph H and a function f(n), the Ramsey–Turán number RT(n,H,f(n)) is the maximum number of edges in an n-vertex H-free graph with independence number at most f(n). For H being a small clique, many results about RT(n,H,f(n)) are known and we focus our attention on H=Ks for s≤13.By applying Szemerédi’s Regularity Lemma, the dependent random choice method and some weighted Turán-type results, we prove that these cliques have the so-called phase transitions when f(n) is around the inverse function of the off-diagonal Ramsey number of Kr versus a large clique Kn for some r≤s.

Adjacency Graphs of Polyhedral Surfaces

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_5$$\\end{document}, K5,81\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{5,81}$$\\end{document}, or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K4,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{4,4}$$\\end{document}, and K3,5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{3,5}$$\\end{document} can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (Isr. J. Math. 46(1–2), 127–144 (1983)), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n\\log n)$$\\end{document}. From the non-realizability of K5,81\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{5,81}$$\\end{document}, we obtain that any realizable n-vertex graph has O(n9/5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(n^{9/5})$$\\end{document} edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

Planar Turán number of the disjoint union of cycles

The planar Turán number of H, denoted by exP(n,H), is the maximum number of edges in an n-vertex H-free planar graph. The planar Turán number of k≥3 vertex-disjoint union of cycles is a trivial value 3n−6. Lan, Shi and Song determine the exact value of exP(n,2C3). We continue to study planar Turán number of two vertex-disjoint union of cycles and obtain the exact value of exP(n,H), where H is vertex-disjoint union of C3 and C4. The extremal graphs are also characterized. We also improve the lower bound of exP(n,2Ck) when n is sufficiently large.