3-connected Planar Graphs Research Articles

Parity vertex colouring of plane graphs

A proper vertex colouring of a 2-connected plane graph G is a parity vertex colouring if for each face f and each colour c , either no vertex or an odd number of vertices incident with f is coloured with c . The minimum number of colours used in such a colouring of G is denoted by χ p ( G ) . In this paper, we prove that χ p ( G ) ≤ 118 for every 2-connected plane graph G .

On the strong parity chromatic number

A vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs.

A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

Let be a nonseparable plane graph on vertices with at least two edges. Suppose that has outer face and that every 2-vertex-cut of contains at least one vertex of . Let denote the chromatic polynomial of . We show that for all . This result is a corollary of a more general result that for all , where is the multivariate Tutte polynomial of , , for all which are not incident to a vertex of , for all , for all other edges , and , are suitably chosen intervals with .

Binary Labelings for Plane Quadrangulations and their Relatives

Graphs and Algorithms Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs.

Diameter bounds for planar graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds ( 4 ( | V | − 1 ) − | E | ) / 3 and 4 | V | 2 / 3 | E | on the diameter (for connected planar graphs), which are also tight.

Facial non-repetitive edge-coloring of plane graphs

A sequence r1, r2, …, r2n such that ri=rn+ i for all 1≤i≤n is called a repetition. A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non-repetitive if the sequence of colors of its edges is non-repetitive. If G is a plane graph, a facial non-repetitive edge-coloring of G is an edge-coloring such that any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) is non-repetitive. We denote π′f(G) the minimum number of colors of a facial non-repetitive edge-coloring of G. In this article, we show that π′f(G)≤8 for any plane graph G. We also get better upper bounds for π′f(G) in the cases when G is a tree, a plane triangulation, a simple 3-connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 38–48, 2010

Distributed computation of virtual coordinates for greedy routing in sensor networks

Sensor networks are emerging as a paradigm for future computing, but pose a number of challenges in the fields of networking and distributed computation. One challenge is to devise a greedy routing protocol—one that routes messages through the network using only information available at a node or its neighbors. Modeling the connectivity graph of a sensor network as a 3-connected planar graph, we describe how to compute on the network in a distributed and local manner a special geometric embedding of the graph. This embedding supports a geometric routing protocol called “greedy routing” based on the “virtual” coordinates of the nodes derived from the embedding.

Computing large matchings fast

In this article we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the best-known algorithm for computing maximum matchings in general graphs, which runs in O (√ nm ) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3, the bounds we achieve are known to be best possible. We also investigate graphs with block trees of bounded degree, where the d -block tree is the adjacency graph of the d -connected components of the given graph. In 3-regular graphs and 3-connected planar graphs with bounded-degree 2- and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5 n − 6 . Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11 n − 18 and 3 n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.

Small Grid Embeddings of 3-Polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.

A generalized greedy routing algorithm for 2-connected graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. One main drawback of this approach is that the coordinates of the virtual locations require Ω ( n log n ) bits to represent, which makes this scheme infeasible in some applications. The essence of the geometric routing is the following: When an origin vertex u wants to send a message to a destination vertex w , it forwards the message to a neighbor t , solely based on the location information of u , w and all neighbors of u . In the greedy routing scheme, the decision is based on decreasing distance. For this idea to work, however, the decision needs not be based on decreasing distance. As long as the decision is made locally, this scheme will work fine. In this paper, we introduce a version of greedy routing which we call generalized greedy routing algorithm. Instead of relying on decreasing distance, a generalized greedy routing algorithm uses other criteria to determine routing paths, solely based on local information. We present simple generalized greedy routing algorithms based on s t -coordinates (consisting of two integers between 0 and n − 1 ), which are derived from an s t -orientation of a 2-connected plane graph. We also generalize this result to arbitrary trees. Both algorithms are natural and simple to be implemented.

Looseness of Plane Graphs

A face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G* increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.

Sharp bounds for the generalized connectivity [formula omitted

Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n . For a set S of k vertices of G , let κ ( S ) denote the maximum number ℓ of edge-disjoint trees T 1 , T 2 , … , T ℓ in G such that V ( T i ) ∩ V ( T j ) = S for every pair i , j of distinct integers with 1 ≤ i , j ≤ ℓ . A collection { T 1 , T 2 , … , T ℓ } of trees in G with this property is called an internally disjoint set of trees connecting S . Chartrand et al. generalized the concept of connectivity as follows: The k - connectivity, denoted by κ k ( G ) , of G is defined by κ k ( G ) = min { κ ( S ) } , where the minimum is taken over all k -subsets S of V ( G ) . Thus κ 2 ( G ) = κ ( G ) , where κ ( G ) is the connectivity of G . For general k , the investigation of κ k ( G ) is very difficult. We therefore focus on the investigation on κ 3 ( G ) in this paper. We study the relation between the connectivity and the 3 -connectivity of a graph. First we give sharp upper and lower bounds of κ 3 ( G ) for general graphs G , and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if G is a connected planar graph, then κ ( G ) − 1 ≤ κ 3 ( G ) ≤ κ ( G ) , and give some classes of graphs which attain the bounds. In the end we give an algorithm to determine κ 3 ( G ) for general graphs G . This algorithm runs in a polynomial time for graphs with a fixed value of connectivity, which implies that the problem of determining κ 3 ( G ) for graphs with a small minimum degree or connectivity can be solved in polynomial time, in particular, the problem whether κ ( G ) = κ 3 ( G ) for a planar graph G can be solved in polynomial time.

Enumerative Properties of Rooted Circuit Maps

In 1966, Barnette introduced a set of graphs, called circuit graphs, which are obtained from 3-connected planar graphs by deleting a vertex. Circuit graphs and 3-connected planar graphs share many interesting properties which are not satisfied by general 2-connected planar graphs. Circuit graphs have nice closure properties which make them easier to deal with than 3-connected planar graphs for studying some graph-theoretic properties. In this paper, we study some enumerative properties of circuit graphs. For enumeration purpose, we define rooted circuit maps and compare the number of rooted circuit maps with those of rooted 2-connected planar maps and rooted 3-connected planar maps.

A note on 3-connected cubic planar graphs

The length of a longest cycle in a graph G is called the circumference of G and is denoted by c ( G ) . Let c ( n ) = min { c ( G ) : G is a 3-connected cubic planar graph of order n } . Tait conjectured in 1884 that c ( n ) = n , and Tutte disproved this in 1946 by showing that c ( n ) ≤ n − 1 for n = 46 . We prove that the inequality c ( n ) ≤ n − n + 49 4 + 5 2 holds for infinitely many integers n . The exact value of c ( n ) is unknown.

On Barnette’s conjecture

Barnette’s conjecture is the statement that every cubic 3-connected bipartite planar graph is Hamiltonian. We show that if such a graph has a 2-factor F which consists only of facial 4-cycles, then the following properties are satisfied: (1) If an edge is chosen on a face and this edge is in F , there is a Hamilton cycle containing all other edges of this face. (2) If any face is chosen, there is a Hamilton cycle which avoids all edges of this face which are not in F . (3) If any two edges are chosen on the same face, there is a Hamilton cycle through one and avoiding the other. (4) If any two edges are chosen which are an even distance apart on the same face, there is a Hamilton cycle which avoids both.

A note on cyclic chromatic number

A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆ ∗ + 2 for any 3-connected plane graph G with maximum face degree ∆. It is known that the conjecture holds true for ∆ ≤ 4 and ∆ ≥ 18. The validity of the conjecture is proved in the paper for some special classes of planar graphs.

Minimum face-spanning subgraphs of plane graphs

A plane graph is a planar graph with a fixed embedding. Let G = (V, E) be an edge weighted connected plane graph, where V and E are the set of vertices and edges, respectively. Let F be the set of faces of G. For each edge e ∊ E, let w(e) ≥ 0 be the weight of the edge e of G. A face-spanning subgraph of G is a connected subgraph H induced by a set of edges S ⊆ E such that the vertex set of H contains at least one vertex from the boundary of each face f ∊ F of G. A minimum face-spanning subgraph H of G is a face-spanning subgraph of G, where is minimum. In this paper we consider the problem of finding a minimum face-spanning subgraph of a plane graph and deal with the following problem which we call “the face-spanning subgraph problem”: “Is there any face-spanning subgraph H of G such that , for a positive real number b?”. We prove that the face-spanning subgraph problem of a plane graph is NP-complete, which implies that it is unlikely to have a polynomial time algorithm for finding a minimum face-spanning subgraph of a plane graph. In this paper, we consider a variation of the face-spanning subgraph problem called “minimum-vertex face-spanning subgraph problem” where the objective is to minimize the number of vertices instead of edge cost and we prove that the minimum-vertex face-spanning subgraph problem is also NP-complete. We also present approximation algorithms for both the problems. We calculate the approximation ratios and time complexities for both the algorithms. In this paper, we also present a linear time algorithm for finding a minimal face-spanning subgraph of a plane graph. We calculate the upper bound and lower bound on the number of vertices for a minimal face-spanning subgraph. We show that the upper bound and the lower bound are tight. Note that the graphs being dealt with in the paper are all undirected.

Greedy Drawings of Triangulations

Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak (Theor. Comput. Sci. 344(1):3–14, 2005) came up with the following conjecture: Any 3-connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s=v 1,v 2,…,v k<=t in a drawing is said to be distance decreasing if ‖v i−t‖<‖v i−1−t‖,2≤i≤k where ‖…‖ denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the Knaster–Kuratowski–Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.

A min–max theorem for plane bipartite graphs

We consider a partitioning problem, defined for bipartite and 2-connected plane graphs, where each node should be covered exactly once by either an edge or by a cycle surrounding a face. The objective is to maximize the number of face boundaries in the partition. This problem arises in mathematical chemistry in the computation of the Clar number of hexagonal systems. In this paper we establish that a certain minimum weight covering problem of faces by cuts is a strong dual of the partitioning problem. Our proof relies on network flow and linear programming duality arguments, and settles a conjecture formulated by Hansen and Zheng in the context of hexagonal systems [P. Hansen, M. Zheng, Upper Bounds for the Clar Number of Benzenoid Hydrocarbons, Journal of the Chemical Society, Faraday Transactions 88 (1992) 1621–1625].