Planar Graphs Research Articles

Planar Graphs with the Maximum Number of Induced 4-Cycles or 5-Cycles

For large n we determine exactly the maximum numbers of induced C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_4$$\\end{document} and C5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_5$$\\end{document} subgraphs that a planar graph on n vertices can contain. We show that K2,n-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{2,n-2}$$\\end{document} uniquely achieves this maximum in the C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_4$$\\end{document} case, and we identify the graphs which achieve the maximum in the C5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_5$$\\end{document} case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.

The polyomino graphs whose resonance graphs have a 1-degree vertex

A polyomino graph is a 2-connected plane graph such that each interior face is a square of side length one. The resonance graph of a polyomino graph P is a graph in which the vertices are the 1-factors of P and two vertices are adjacent provided their corresponding two 1-factors differ in exactly one unite square. In this paper, we determine the polyomino graphs whose resonance graph has a 1-degree vertex by constructing a 4-coordinate system. As consequences we determine the polyomino graphs with a forcing edge and an anti-forcing edge respectively.

Partitioning planar graphs into bounded degree forests

An (Fd1,Fd2)-partition of a graph G is a partition of V(G) into two subsets V1 and V2 such that G[Vi] is a forest with maximum degree at most di for i=1,2. In this paper we show that every planar graph without 4-cycles and 6-cycles has an (F5,F5)-partition. This improves a result by Huang, Huang and Lv in 2023, which says that G has an (F2,F)-partition.

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette N in which the set of colours of edges that are incident to each vertex is an interval. The interval colouring thickness of a graph G is the minimum k such that G can be edge-decomposed into k interval colourable graphs. We show that θ(n), the maximum interval colouring thickness of an n-vertex graph, satisfies θ(n)=Ω(log(n)/loglog(n)) and θ(n)⩽n5/6+o(1), which improves on the trivial lower bound and the upper bound given by the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an n-vertex planar graph uses at most 3n/2−2 colours.

Upward planarity testing of biconnected outerplanar DAGs solves partition

We show an O(n)-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to n/2 to the problem of testing whether an n-vertex biconnected outerplanar DAG admits an upward planar drawing. This constitutes the first barrier to the existence of efficient algorithms for testing the upward planarity of DAGs with no large triconnected minor.We also show a result in the opposite direction. Suppose that partitioning a multiset of positive integers into two multisets so that the sum of the integers in each multiset is n/2 can be solved in f(n) time. Let G be an n-vertex biconnected outerplanar DAG and e be an edge incident to the outer face of an outerplanar drawing of G. Then it can be tested in O(f(n)) time whether G admits an upward planar drawing with e on the outer face.

Bounding clique size in squares of planar graphs

Wegner conjectured that if G is a planar graph with maximum degree Δ≥8, then χ(G2)≤32Δ+1. This problem has received much attention, but remains open for all Δ≥8. Here we prove an analogous bound on ω(G2): If G is a plane graph with Δ(G)≥36, then ω(G2)≤⌊32Δ(G)⌋+1. In fact, this is a corollary of the following lemma, which is our main result. If G is a plane graph with Δ(G)≥19 and S is a maximal clique in G2 with |S|≥Δ(G)+20, then there exist x,y,z∈V(G) such that S={w:|N[w]∩{x,y,z}|≥2}.

On planarity of graphs in homotopy type theory

AbstractIn this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

Tropical graph curves

We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as tropical graph curves. Roughly speaking, the tropical graph curve associated to $G$ , whose genus is $g$ , is an arrangement of $2g-2$ lines in tropical projective space that contains $G$ (more precisely, the topological space associated to $G$ ) as a deformation retract. We show the existence of tropical graph curves when the underlying graph is a three-regular, three-vertex-connected planar graph. Our method involves explicitly constructing an arrangement of lines in projective space, i.e. a graph curve whose tropicalization yields the corresponding tropical graph curve and in this case, solves a topological version of the tropical lifting problem associated to canonically embedded graph curves. We also show that the set of tropical graph curves that we construct are connected via certain local operations. These local operations are inspired by Steinitz’ theorem in polytope theory.

Locating-dominating number of certain infinite families of convex polytopes with applications

A convex hull of finitely many points in the Euclidean space Rd is known as a convex polytope. Graphically, they are planar graphs i.e. embeddable on R2. Minimum dominating sets possess diverse applications in computer science and engineering. Locating-dominating sets are a natural extension of dominating sets. Studying minimizing locating-dominating sets of convex polytopes reveal interesting distance-dominating related topological properties of these geometrical planar graphs. In this paper, exact value of the locating-dominating number is shown for one infinite family of convex polytopes. Moreover, tight upper bounds on γl−d are shown for two more infinite families. Tightness in the upper bounds is shown by employing an updated integer linear programming (ILP) model for the locating-dominating number γl−d of a fixed graph. Results are explained with help of some examples. The second part of the paper solves an open problem in Khan (2023) [28] which asks to find a domination-related parameter which delivers a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the locating-dominating number γl−d delivers such a strong prediction potential. The paper is concluded with putting forward some open problems in this area.

Choosing Friends in Everyday Life by using Graph Theory

Graph theory is playing vital role in almost every field of our routine life. You make a conference call with your friends by using vertices (yourself and your friends) and edges (network connection). You construct a printed grid floor with different faces in your home by the help of graph theory. Authors in this study are using labelling of graphs and applying it in choosing best friends around you. The helping graphs in this article will be plane graphs which will be labelling under k − labelling M of kind ( λ , μ , ν ) . This study can be applied in many fields of everyday life.

On Decompositions of \(\mathbf{K_{18k}}\) and \(\mathbf{K_{18k+1}}\) into Certain Connected Bicyclic Graphs with Nine Edges

We show that connected, bicyclic graphs on nine edges with at least one cycle other than C 3 decompose the complete graphs K 18 k and K 18 k + 1 , for k ≥ 1 , when the necessary conditions allow for such a decomposition. This complements previous results by Freyberg, Froncek, Jeffries, Jensen, and Sailstad on connected bicyclic triangular graphs.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

OPTICAL ART OF A PLANAR IDEMPOTENT DIVISOR GRAPH OF COMMUTATIVE RING

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper.

Models of Klein Surface Obstruction Graphs

The task of researching the structure of graphs of given connectivity, which are obstructions for a given surface of non-oriented kind, and building their models, from which obstruction graphs are formed by removing or compressing a set of edges, is considered. The issue of edge coverage of an obstruction graph of a given kind with a minimum number of quasi-stars with centers – planar graphs that have given sets of points and all edges are significant with respect to the reachability number 2 on the Euclidean plane and has reachability on the projective plane or Klein surface, is considered. K4, K2,3 or a degenerate graph. The task of researching the structure of graphs of undirected kind was considered [4–6]. In [7], the set of minors for the projective plane was compressed to 12 basic minors using the method of relative components, and a set of 62 minors of the Klein surface was constructed. To do this, we considered all non-isomorphic minimal embeddings of each of the basic minors and found the set of all different pairs of vertices that are reachable on the projective plane during the operations of removing or compressing an arbitrary edge of this graph to a point, then a pair of non-adjacent graph vertices was attached to the selected pair of points. In [8], the number of 2-connected obstruction graphs for the Klein surface was calculated, part of the diagrams of these graphs is given in [10]. Note that the following definition of the cell distance is similar to that in [11].Our approach, as a continuation of [9], will consist in finding the edge covering of an obstruction graph of a given kind by the minimum number of subgraphs of the covering from the number of quasi-stars with centers - graphs with essential edges relative to the number of reachability or nonorientable genus during compression to a point or removal operations edges relative to a given set of points with reachability number 2 relative to the Euclidean plane and reachable on projective planes or Klein surfaces, for example, these are subsets of the set of points of graphs K4, K2,3, K5\e, Kr, r >= 2, or graph-obstructions of the projective plane. We also found the necessary conditions for constructing obstruction graphs for the Klein surface by identifying pairs of center points and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. Hypothetically, a graph-obstruction of a given nonorientable genus has the form of a cylindrical surface with n, n >= 2, disks-bases and a side part, which can have common sets of points on the boundaries and on which are embedded, at least in part, the graph-centers of quasi-stars having a given set of reachability points 2 on the Euclidean plane, and on the side surface there are hanging edges that intersect on the plane and are inserted without crossing with the help of Mobius strips glued to the side surface. At the same time, the edges will have at least two nesting options in the side part of the cylindrical surface, but no more than the number of glued Mobius strips, thanks to which each hanging edge will nest on the Mobius strip, either with only one edge or with two adjacent edges. We have found the necessary conditions for constructing models of obstruction graphs for the Klein surface by identifying pairs of centers and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. The main result: statements 1, 2, 3 and the algorithm for constructing models of 3-connected graph-obstructions of the Klein surface. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.

The T1,2-free planar graphs whose second largest eigenvalue does not exceed 1

The problem of characterizing graphs whose second largest eigenvalue does not exceed a comparatively small constant has attracted a great deal of attention in the last decades. Let T1,2 be the tree of diameter 3 with exactly 3 pendent vertices. In this paper, all connected T1,2-free planar graphs whose second largest eigenvalue does not exceed 1 are determined.

Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology

In the Multiagent Path Finding (MAPF for short) problem, we focus on efficiently finding non-colliding paths for a set of k agents on a given graph G, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule l, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our Fixed-Parameter Tractability (FPT) results. We show that MAPF is W[1]-hard with respect to k (even if k is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan l are fixed constants. On the positive side, we show an FPT algorithm for k+l. As we continue, the structure of G comes into play. We give an FPT algorithm for parameter k plus the diameter of the graph G. The MAPF problem is W[1]-hard for cliquewidth of G plus l while it is FPT for treewidth of G plus l.

A note on the list vertex-arboricity of IC-planar graphs

The vertex-arboricity a(G) of a graph G is the minimum number of colors required to color the vertices of G such that no cycle is monochromatic. The list vertex-arboricity al(G) of G is the list-coloring version of this invariant. It was known that every planar graph G has al(G)≤3. In this paper, we extend this result by showing that every graph G drawn in the plane so that the distance between every pair of crossings is at least 2 has al(G)≤3.

Cyclic edge and cyclic vertex connectivity of [formula omitted]-fullerene graphs

Cyclic vertex connectivity cκ and cyclic edge connectivity cλ are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that cκ=cλ for (4,5,6)-fullerene graphs except for four exceptional graphs with order less than 16. We also give O(ν)-algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4,5,6)-fullerene graphs.

Generation and new infinite families of K2-hypohamiltonian graphs

We present an algorithm which can generate all pairwise non-isomorphic K2-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for K2-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which K2-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite K2-hypohamiltonian graphs. Furthermore, we describe a new operation for creating K2-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar K2-hypohamiltonian graph of order n for every integer n≥134. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence K2-hypohamiltonian graphs with large maximum degree and size.

Independence number and spectral radius of cactus graphs

Four Color Theorem implies that the independence number of every planar graph is at least n/4. A cactus graph is a connected planar graph whose block is either an edge or a cycle. In this paper, the lower bound for independence number of cactus graphs is improved to ⌈n/3⌉. Moreover, the maximum spectral radius of cactus graphs with fixed independence number is determined.