Planar Graphs Research Articles

The maximum number of pentagons in a planar graph

AbstractIn 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an ‐vertex planar graph. They precisely determined the maximum number of triangles and four‐cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the ‐vertex, planar graphs with the maximum number of pentagons.

Type-II Apollonian network: More robust and more efficient Apollonian network

The family of planar graphs is a particularly important family and models many networks including the layout of printed circuits. The widely-known Apollonian packing process has been used as guideline to create the typical Apollonian network with planarity. In this paper, we propose a new principled framework based on the Apollonian packing process to generate model as complex network, and obtain a family of new networks called Type-II Apollonian network At. While our network and the typical Apollonian network are maximal planar, the former turns out to be Hamiltonian and Eulerian, however, the latter is not. Then, we in-depth study some fundamental structural properties on network At, and verify that network At is sparse, has scale-free feature and small-world property, and exhibits disassortative mixing structure. Next, we derive the asymptotic solution of the spanning tree entropy of network At by designing an effective algorithm, which suggests that Type-II Apollonian network is more robust to a random removal of edges than the typical Apollonian network. Additionally, we study trapping problem on network At, and use average trapping time as metric to show that Type-II Apollonian network At has more efficient underlying structure for fast information diffusion than the typical Apollonian network.

Planar #CSP Equality Corresponds to Quantum Isomorphism – A Holant Viewpoint

Recently, Mančinska and Roberson proved that two graphs G and \(G^{\prime }\) are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix \(\mathcal {U}\) defining the quantum isomorphism. Due to the noncommutativity of \(\mathcal {U}\) ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group \(Qut(\mathcal {F})\) of a set \(\mathcal {F}\) of constraint functions/tensors and characterize the intertwiners of \(Qut(\mathcal {F})\) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.

Splitting Plane Graphs to Outerplanarity

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is NP-hard.Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove NP-completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Additionally, we show upper and lower bounds for certain families of maximal planar graphs. Finally, we provide a SAT formulation for the problem, and evaluate it on a small benchmark.

A complexity trichotomy for k-regular asymmetric spin systems with complex edge functions

We prove a complexity trichotomy theorem for a class of partition functions over k-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: For every parameter setting in C for the spin system, the partition function is either (1) computable in polynomial time for every graph, or (2) #P-hard for general graphs but computable in polynomial time for planar graphs, or (3) #P-hard even for planar graphs.

Fuzzy Outerplanar Graphs and Its Applications

The concept of a crisp graph is essential in the study of outerplanar graphs because outerplanar graphs are a unique type of planar graphs containing special characteristics. One of the core concepts of crisp graphs, the notion of a subgraph, is utilized primarily to solve difficult data problems. In this paper, researching outerplanar graphs in fuzzy graphs is the main aim of the work. By allowing edges connecting vertices to have varying levels of membership or fuzziness, a fuzzy graph is a mathematical framework that develops on the concept of crisp graphs. Because of their capacity to describe unclear or imprecise relationships between items, fuzzy outerplanar graphs might have potential applications. The subgraphs of fuzzy outerplanar graphs are revealed by eliminating specific vertices or edges from the fuzzy graphs. Furthermore, maximum, and maximal fuzzy outerplanar subgraphs defined for both vertex and edge deletion are examined using examples. Some of the connections between the concepts discussed are represented as theorems and conclusions. An application for the layout of a bypass road is discussed using the proposed concepts.

The Linear Arboricity of IC-planar Graphs

The linear arboricity of a graph is the minimum number of linear forests that partition the edges of . In 1981, Akiyama, Exoo and Harary conjectured that for any simple graph . A graph is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. An IC-planar graph is a 1-planar graph satisfying the condition that each vertex is incident with at most one crossing edge. It is shown in this paper that every IC-planar graph with maximum degree has .

Rectangle-of-Influence Drawings of Five-Connected Plane Graphs

Rectangle-of-Influence Drawings of Five-Connected Plane Graphs

List Equitable Coloring of Planar Graphs without $4$- and $6$-Cycles when $\Delta(G)=5$

A graph $G$ is $k$ list equitably colorable, if for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. In 2009, Li and Bu obtained that for planar graph $G$, if $\Delta(G)\geq6$ and without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable. In order to further prove the conjecture of list equitable coloring, in this paper, we focus on planar graph with $\Delta(G)=5$, and prove that if $G$ is a planar graph without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable.

Dissecting polytopes: Landau singularities and asymptotic expansions in 2 → 2 scattering

Parametric representations of Feynman integrals have a key property: many, frequently all, of the Landau singularities appear as endpoint divergences. This leads to a geometric interpretation of the singularities as faces of Newton polytopes, which facilitates algorithmic evaluation by sector decomposition and asymptotic expansion by the method of regions. Here we identify cases where some singularities appear instead as pinches in parametric space for general kinematics, and we then extend the applicability of sector decomposition and the method of regions algorithms to such integrals, by dissecting the Newton polytope on the singular locus. We focus on 2 → 2 massless scattering, where we show that pinches in parameter space occur starting from three loops in particular nonplanar graphs due to cancellation between terms of opposite sign in the second Symanzik polynomial. While the affected integrals cannot be evaluated by standard sector decomposition, we show how they can be computed by first linearising the graph polynomial and then splitting the integration domain at the singularity, so as to turn it into an endpoint divergence. Furthermore, we demonstrate that obtaining the correct asymptotic expansion of such integrals by the method of regions requires the introduction of new regions, which can be systematically identified as facets of the dissected polytope. In certain instances, these hidden regions exclusively govern the leading power behaviour of the integral. In momentum space, we find that in the on-shell expansion for wide-angle scattering the new regions are characterised by having two or more connected hard subgraphs, while in the Regge limit they are characterised by Glauber modes.

The Planar Turán Number of {K4,C5}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{K_4,C_5\\}$$\\end{document} and {K4,C6}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{K_4,C_6\\}$$\\end{document}

Let H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} be a set of graphs. The planar Turán number, exP(n,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,\\mathcal {H})$$\\end{document}, is the maximum number of edges in an n-vertex planar graph which does not contain any member of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} as a subgraph. When H={H}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}=\\{H\\}$$\\end{document} has only one element, we usually write exP(n,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,H)$$\\end{document} instead. The study of extremal planar graphs was initiated by Dowden (J Graph Theory 83(3):213–230, 2016). He obtained sharp upper bounds for both exP(n,C5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,C_5)$$\\end{document} and exP(n,K4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,K_4)$$\\end{document}. Later on, sharp upper bounds were proved for exP(n,C6)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,C_6)$$\\end{document} and exP(n,C7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,C_7)$$\\end{document}. In this paper, we show that exP(n,{K4,C5})≤157(n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,\\{K_4,C_5\\})\\le {15\\over 7}(n-2)$$\\end{document} and exP(n,{K4,C6})≤73(n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ex_\\mathcal {P}(n,\\{K_4,C_6\\})\\le {7\\over 3}(n-2)$$\\end{document}. We also give constructions which show the bounds are sharp for infinitely many n.

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

Odd facial total-coloring of maximal plane and outerplane graphs

Odd facial total-coloring of maximal plane and outerplane graphs

Baxter’s Theorem: A Connection Between Combinatorics and Topology

Summary We provide two proofs of a theorem due to Baxter on the four-coloring of a cubic planar graph. We then consider an n-dimensional version of Baxter’s theorem and relate it to Sperner’s lemma.

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.

Proper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s

AbstractWe call a proper edge coloring of a graph a B‐coloring if every 4‐cycle of is colored with four different colors. Let denote the smallest number of colors needed for a B‐coloring of . Motivated by earlier papers on B‐colorings, here we consider for planar and outerplanar graphs in terms of the maximum degree . We prove that for planar graphs, for bipartite planar graphs, and for outerplanar graphs with . We conjecture that, for sufficiently large, for planar , and for outerplanar .

A face cover perspective to \(\ell_{1}\) embeddings of planar graphs

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into \(\ell_{1}\) with constant distortion. However, given an \(n\) -vertex weighted planar graph, the best upper bound on the distortion is only \(O(\sqrt{\log n})\) , by Rao [SoCG99]. In this paper we study the case where there is a set \(K\) of terminals, and the goal is to embed only the terminals into \(\ell_{1}\) with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into \(\ell_{1}\) . The more general case, where the set of terminals can be covered by \(\gamma\) faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of \(O(\log\gamma)\) by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to \(O(\sqrt{\log\gamma})\) . Since every planar graph has at most \(O(n)\) faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into \(\ell_{1}\) . Therefore, our result provides a polynomial time \(O(\sqrt{\log\gamma})\) -approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by \(\gamma\) faces.

On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

Computing planar orthogonal drawings with the minimum number of bends is one of the most studied topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia in SIAM J Comput 31(2):601–625, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number b of bends, the number k of vertices of degree at most two, and the treewidth tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{tw}$$\\end{document} of the input graph (Di Giacomo et al. in J Comput Syst Sci 125:129–148, 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by b+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b+k$$\\end{document}. As a consequence, rectilinear planarity testing lies in FPT parameterized by the number of vertices of degree at most two. We also prove that our choice of parameters is minimal, as deciding if an orthogonal drawing with at most b bends exists is already NP-hard when k is zero (i.e., the problem is para-NP-hard parameterized in k); hence, there is neither an FPT nor an XP algorithm parameterized only by the parameter k (unless P = NP). In addition, we prove that the problem is W[1]-hard parameterized by k+tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+\ extsf{tw}$$\\end{document}, complementing a recent result (Jansen et al. in Upward and orthogonal planarity are W[1]-hard parameterized by treewidth. CoRR, abs/2309.01264, 2023; in: Bekos MA, Chimani M (eds) Graph Drawing and Network Visualization, vol 14466, Springer, Cham, pp 203–217, 2023) that shows W[1]-hardness for the parameterization b+tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b+\ extsf{tw}$$\\end{document}. As a consequence, we are able to trace a clear parameterized tractability landscape for the bend-minimum orthogonal planarity problem with respect to the three parameters b, k, and tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{tw}$$\\end{document}.

A special graph for the connected metric dimension of graphs

Given a connected graph G=(V, E), let d(x, y) represent the separation between x and y at its vertices. If each vertex in a collection B is uniquely identified by its vector of distances to the vertices in B, then that set of vertices resolves a graph G. A metric dimension of G is represented by dim(G) and is the smallest cardinality of a resolving set of G. If the subgraph B- induced by B is a nontrivial connected subgraph of G, then a resolving set B of G is connected. The metric dimension of G is the cardinality of the minimal resolving set, while the connected metric dimension of G is the cardinality of the smallest connected resolving set. The connected metric dimension of the knots graph, whitehead link graph and jewel graph are determined in this study. Finally, we derive the explicit formulas for the triangular book graph, quadrilateral book graph and crystal planar map.

RAINBOW VERTEX CONNECTION NUMBER OF BULL GRAPH, NET GRAPH, TRIANGULAR LADDER GRAPH, AND COMPOSITION GRAPH (P_n [P_1

The rainbow connection was first introduced by Chartrand in 2006 and then in 2009 Krivelevich and Yuster first time introduced the rainbow vertex connection. Let graph be a connected graph. The rainbow vertex-connection is the assignment of color to the vertices of a graph , if every vertex on the graph is connected by a path graph that has interior vertices in different colors. The minimum number of colors from the rainbow vertex coloring in the graph is called rainbow vertex connection number which is denoted . The results of the research are the rainbow vertex connection number of bull graph, net graph, triangular ladder graph, and graph composition (Pn[P1]).