3-connected Planar Graphs Research Articles

Strong Isoperimetric Inequalities and Combinatorial Curvatures on Multiply Connected Planar Graphs

The main focus of this paper is on hyperbolic properties of multiply connected planar graphs (planar graphs with multiple ends), and in the course we study some problematic phenomena of planar graphs caused by the existence of multiple (or sometimes infinite) ends. Specifically, in the first part of the paper we examine strong isoperimetric inequalities on a multiply connected planar graph G and its dual graph $$G^*$$Gź, and prove that G satisfies a strong isoperimetric inequality if and only if $$G^*$$Gź has the same property, provided that G is either normal or finitely connected and we choose an appropriate notion for strong isoperimetric inequalities. In the second part we study a planar graph G on which negative curvatures uniformly dominate positive curvatures, and give a criterion that guarantees a strong isoperimetric inequality on G. Our criterion is useful in that it can be applied to a graph containing a long and slim subgraph with nonnegative combinatorial curvatures.

Facial list colourings of plane graphs

Let G=(V,E,F) be a connected plane graph, with vertex set V, edge set E, and face set F. For X∈{V,E,F,V∪E,V∪F,E∪F,V∪E∪F}, two distinct elements of X are facially adjacent in G if they are incident elements, adjacent vertices, adjacent faces, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A list k-colouring is facial with respect to X if there is a list k-colouring of elements of X such that facially adjacent elements of X receive different colours. We prove that every plane graph G=(V,E,F) has a facial list 4-colouring with respect to X=E, a facial list 6-colouring with respect to X∈{V∪E,E∪F}, and a facial list 8-colouring with respect to X=V∪E∪F. For plane triangulations, each of these results is improved by one and it is tight. These results complete the theorem of Thomassen that every plane graph has a (facial) list 5-colouring with respect to X∈{V,F} and the theorem of Wang and Lih that every simple plane graph has a (facial) list 7-colouring with respect to X=V∪F.

Lower and upper bounds for long induced paths in 3-connected planar graphs

Let G be a 3-connected planar graph with n vertices and let p(G) be the maximum number of vertices of an induced subgraph of G that is a path. Substantially improving previous results, we prove that p(G)≥log⁡n12log⁡log⁡n. To demonstrate the tightness of this bound, we notice that the above inequality implies p(G)∈Ω((log2⁡n)1−ε), where ε is any positive constant smaller than 1, and describe an infinite family of planar graphs for which p(G)∈O(log⁡n). As a byproduct of our research, we prove a result of independent interest: Every 3-connected planar graph with n vertices contains an induced subgraph that is outerplanar and connected and that contains at least n3 vertices. The proofs in the paper are constructive and give rise to O(n)-time algorithms.

On purely tree-colorable planar graphs

A tree-k-coloring of a graph G is a k-coloring of G such that the subgraph induced by the union of any two color classes is a tree. G is purely tree-k-colorable if the chromatic number of G is k and any k-coloring of G is a tree-k-coloring. Xu [16] conjectured that there exist only two purely tree-4-colorable 4-connected maximal planar graphs. In this paper, we construct an infinite family of purely tree-colorable 4-connected maximal planar graphs, called dumbbell-maximal planar graphs, which disprove Xu's conjecture. Moreover, we give the enumeration of dumbbell-maximal planar graphs and propose a conjecture on such graphs. It turns out that the conjecture implies naturally the uniquely 4-colorable planar graph conjecture.

On self-approaching and increasing-chord drawings of 3-connected planar graphs

An $st$-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from $s$ to any point $t'$ on the curve the distance to $t'$ is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path. We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. (GD 2014). Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.

5-Connected Toroidal Graphs are Hamiltonian-Connected

The problem on the Hamiltonicity of graphs is well studied in discrete algorithm and graph theory because of its relation to the traveling salesman problem. Starting with Tutte's result, stating that every 4-connected planar graph is Hamiltonian, several researchers have studied the Hamiltonicity of graphs on surfaces. Extending Tutte's technique, Thomassen proved that every 4-connected planar graph is in fact Hamiltonian-connected, i.e., there is a Hamiltonian path connecting any two prescribed vertices. For graphs on the torus, Thomas and Yu showed that every 5-connected graph on the torus has a Hamiltonian cycle. In this paper, we prove the following result which generalizes Thomas and Yu's result. Every 5-connected graph on the torus is Hamiltonian-connected. Our result is best possible in the sense that we cannot lower the connectivity 5 (i.e., there is a 4-connected graph on the torus which is not Hamiltonian-connected). Moreover, our proof is constructive in a sense that it gives rise to a polynomi...

A new proof that 4-connected planar graphs are Hamiltonian-connected

A new proof that 4-connected planar graphs are Hamiltonian-connected

Bimagic Labeling for Certain Families of Strong Face Graphs

Let G = (V, E, F) be a simple, finite, connected plane graph with V(G) as a vertex set, E(G) an edge set and F(G) set of faces, a labeling of type (1, 1, 0) is a bijection from the vertex set and edge set of a graph G to the set {1, 2, …, |V(G)|+|E(G)|}. This labeling is called super if the set of vertices received the smallest number. The weight of a face w(f) under this labeling is sum of labels of the vertices and edges surrounding this face.

On longest cycles in essentially 4-connected planar graphs

On longest cycles in essentially 4-connected planar graphs

THE ENTIRE FACE IRREGULARITY STRENGTH OF A BOOK WITH POLYGONAL PAGES

A face irregular entire labeling is introduced by Baca et al. recently, as a modification of the well-known vertex irregular and edge irregular total labeling of graphs and the idea of the entire colouring of plane graph. A face irregular entire k-labeling 𝜆:𝑉∪𝐸∪𝐹→{1,2,⋯,𝑘} of a 2-connected plane graph 𝐺=(𝑉,𝐸,𝐹) is a labeling of vertices, edges, and faces of 𝐺 such that for any two different faces 𝑓 and 𝑔, their weights 𝑤𝜆(𝑓) and 𝑤𝜆(𝑓) are distinct. The minimum 𝑘 for which a plane graph 𝐺 has a face irregular entire 𝑘-labeling is called the entire face irregularity strength of 𝐺, denoted by 𝑒𝑓𝑠(𝐺). This paper deals with the entire face irregularity strength of a book with 𝑚 𝑛-polygonal pages, where embedded in a plane as a closed book with 𝑛−sided external face.

Triangles in random cubic planar graphs

In this extended abstract we determine a normal limiting distribution for the number of triangles in a uniformly at random 3-connected cubic planar graph, as well as the precise expectation and variance values. Further comments towards the more complicated problem of studying both the limiting distribution of triangles in random cubic planar graphs, and the (asymptotic) number of triangle-free cubic planar graphs are discussed as well.

On k-greedy routing algorithms

Let G=(V,E) and G′=(V′,E′) be two graphs. A k-inverse-adjacency-preserving mapping ρ from G to G′ is a one-to-many and onto mapping from V to V′ satisfying the following: (1) Each vertex v∈V in G is mapped to a non-empty subset ρ(v)⊂V′ in G′, the cardinality of ρ(v) is at most k; (2) if u≠v, then ρ(u)∩ρ(v)=∅; and (3) for any u′∈ρ(u) and v′∈ρ(v), if (u′,v′)∈E′, then (u,v)∈E. A vertex u′ in ρ(u) is called a virtual location for u.Let ρ be a k-inverse-adjacency-preserving-mapping (k-IAPM for short) from G to G′. Let δ be a greedy drawing of G′ into a metric space M. Consider a message from u to be delivered to v in G. Using the k-IAPM ρ from G to G′, intuitively, one can treat the message to be routed as if it were from one virtual location u′ for u to one virtual location v′ for v, except that all the virtual locations of a vertex u were identified with each other (which can be thought as instantaneously synchronized mirror sites for a particular website, for example). Then a routing path P′ can be computed from u′ to v′ while identifying all virtual locations for any vertex in G. Since ρ inversely preserves adjacency from G′ to G, such a routing path P′ corresponds to exactly one routing path P in G, which connects u to v.In this paper, we formalize the above intuition into a concept which we call k-greedy routing algorithm for a graph G with n vertices, where k refers to the maximum number of virtual locations any vertex of G can have. Using this concept, the result presented in [18] can be rephrased as a 3-greedy routing algorithm for 3-connected plane graphs, where the virtual coordinates used are from 1 to 2n−2. In this paper, we present a 2-greedy routing algorithm for 3-connected plane graphs, where each vertex uses at least one but at most two virtual locations numbered from 1 to 2n−1. For the special case of plane triangulations (in a plane triangulation, every face is a triangle, including the exterior face), the numbers used are further reduced to from 1 to ⌊5n+13⌋. Hence, there are at least ⌈n−13⌉ vertices that use only one virtual location.

Decomposing plane cubic graphs

It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and 3-connected cubic graphs on the projective plane. Our proof provides a polynomial time algorithm to find the decomposition for 3-connected cubic plane graphs.

Good spanning trees in graph drawing

A plane graph is a planar graph with a fixed planar embedding. In this paper we define a special spanning tree of a plane graph which we call a good spanning tree. Not every plane graph has a good spanning tree. We show that every connected planar graph has a planar embedding with a good spanning tree. Using a good spanning tree, we show that every connected planar graph G of n vertices has a straight-line monotone grid drawing on an O(n)×O(n2) grid, and such a drawing can be found in O(n) time. Our results solve two open problems on monotone drawings of planar graphs posed by Angelini et al. Using good spanning trees, we also give simple linear-time algorithms for finding a 2-visibility representation of a connected planar graph G of n vertices on a (2n−1)×(2n−1) grid and for finding a spike-VPG representation of G on a (2n−1)×n grid.

Edge looseness of plane graphs

A face of an edge colored plane graph is called e-loose if the number of colors used on its edges is at least three. The e-looseness of a plane graph G is the minimum positive integer k such that any edge coloring of G with k colors involves an e-loose face. In this paper we determine tight lower and upper bounds for the e-looseness of connected plane graphs. These bounds are expressed by linear polynomials of the number of faces.

Verification on a Given Point Set for a Cubic Plane Graph

Cubic graph, where all vertices have degree three can be associated as 3-regular graph and trivalent graph. Let a point P be the given point set, with a stipulation of n ≥ 4 points in the regular plane such that n is even in general situation. It has been probed that the problem whether there is a straight-line cubic plane graph on P. Algorithm is not known for polynomialtime of a problem. With research on the reduction to the existence of the convex hull of P along with the certain diagonals of the boundary cycle, the polynomial-time algorithm is proposed that checks for 2-connected cubic plane graphs. The algorithm is constructive and runs in time complexity of O (n3). It is also shown that which graph structure can be expected when there is a cubic plane graph on P; e. g., a cubic plane graph on P implies a connected cubic plane graph on P, and a 2-connected cubic plane graph on P implies a 2-connected cubic plane graph on P that contains the boundary cycle of P.

Note on 3-paths in plane graphs of girth 4

An (i,j,k)-path is a path on three vertices u, v and w in this order with deg(u)≤i, deg(v)≤j, and deg(w)≤k. In this paper, we prove that every connected plane graph of girth 4 and minimum degree at least 2 has at least one of the following: a (2,∞,2)-path, a (2,7,3)-path, a (3,5,3)-path, a (4,2,5)-path, or a (4,3,4)-path. Moreover, no parameter of this description can be improved. Our result supplements recent results concerning the existence of specific 3-paths in plane graphs.

Wheel-free planar graphs

A wheel is a graph formed by a chordless cycle C and a vertex u not in C that has at least three neighbors in C. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a line graph or has a clique cutset. We prove that every planar graph that does not contain a wheel as an induced subgraph is 3-colorable.

Backbone coloring of planar graphs for [formula omitted]-free or [formula omitted]-free

Let G=(V,E) be a graph and H be a spanning subgraph of G. A backbone-k-coloring of (G,H) is a mapping φ: V(G)→{1,2,⋅⋅⋅,k} such that |φ(u)−φ(v)|≥2 if uv∈E(H) and |φ(u)−φ(v)|≥1 if uv∈E(G)\\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring, denoted by χb(G,H). In this paper, we prove that if G is a connected planar graph that is C8-free or C9-free and without adjacent 4-cycles, then there exists a spanning tree T of G such that χb(G,T)≤4.

Maximum 4-Degenerate Subgraph of a Planar Graph

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a $4$-degenerate induced subgraph containing at least $ (38 - d)/36$ of its vertices. This shows that every planar graph of order $n$ has a $4$-degenerate induced subgraph of order more than $8/9 \cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.