3-regular Planar Graphs Research Articles

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

Coloring circle arrangements: New 4-chromatic planar graphs

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of (pseudo-)circle arrangements.In this paper the conjecture is verified for △-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps △-saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This corona construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture.We further investigate fractional colorings. It is shown that the arrangement graph of every arrangement A of pairwise intersecting pseudocircles is “close” to being 3-colorable. More precisely, the fractional chromatic number χf(A) of the arrangement graph is bounded from above by χf(A)≤3+O(1n), where n is the number of pseudocircles of A. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019).

On the hull number on cycle convexity of graphs

In this work, we study the parameter hull number in a recently defined graph convexity called Cycle Convexity, whose definition is motivated by related notions in Knot Theory.For a graph G=(V,E), define the interval function in the Cycle Convexity as Icc(S)=S∪{v∈V(G)|there is a cycle C in G such that V(C)∖S={v}}, for every S⊆V(G). We say that S⊆V(G) is convex if Icc(S)=S. The convex hull of S⊆V(G), denoted by Hull(S), is the inclusion-wise minimal convex set S′ such that S⊆S′. A set S⊆V(G) is called a hull set if Hull(S)=V(G). The hull number of G in the cycle convexity, denoted by hncc(G), is the cardinality of a smallest hull set of G.We first focus on the class of planar graphs, as the main motivation for the definition of hncc(G) stems from Knot Theory and occurs when G is a 4-regular planar graph. We prove that: the hull number of a 4-regular planar graph is at most half of its number of vertices and that such bound is tight; and that deciding whether hncc(G)≤k, provided a positive integer k and a planar graph G, is an NP-complete problem.On the positive side, we present polynomial-time algorithms to compute the hull number in the cycle convexity of chordal graphs, P4-sparse graphs, and grids.

Anti-Kekulé number of the {(3, 4), 4}-fullerene.

A {(3,4),4}-fullerene graph G is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. An edge subset S of G is called an anti-Kekulé set, if G - S is a connected subgraph without perfect matchings. The anti-Kekulé number of G is the smallest cardinality of anti-Kekulé sets and is denoted by . In this paper, we show that ; at the same time, we determine that the {(3, 4), 4}-fullerene graph with anti-Kekulé number 4 consists of two kinds of graphs: one of which is the graph consisting of the tubular graph , where Q n is composed of concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Figure 2 for the graph Q n ); and the other is the graph , which contains four diamonds D 1, D 2, D 3, and D 4, where each diamond consists of two adjacent triangles with a common edge such that four edges e 1, e 2, e 3, and e 4 form a matching (see Figure 7D for the four diamonds D 1 - D 4). As a consequence, we prove that if , then ; moreover, if , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.

Enumeration of labelled 4-regular planar graphs II: Asymptotics

This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number gn of labelled 4-regular planar graphs on n vertices. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

On reconfigurability of target sets

We study the problem of deciding reconfigurability of target sets of a graph. Given a graph G with vertex thresholds τ, consider a dynamic process in which vertex v becomes activated once at least τ(v) of its neighbors are activated. A vertex set S is called a target set if all vertices of G would be activated when initially activating vertices of S. In the Target Set Reconfiguration problem, given two target sets X and Y of the same size, we are required to determine whether X can be transformed into Y by repeatedly swapping one vertex in the current set with another vertex not in the current set preserving every intermediate set as a target set. In this paper, we investigate the complexity of Target Set Reconfiguration in restricted cases. On the hardness side, we prove that Target Set Reconfiguration is PSPACE-complete on bipartite planar graphs of degree 3 and 4 and of threshold 2, bipartite 3-regular graphs and planar 3-regular graphs of threshold 1 and 2, and split graphs, which is in contrast to the fact that a special case called Vertex Cover Reconfiguration is in P for the last graph class. On the positive side, we present a polynomial-time algorithm for Target Set Reconfiguration on graphs of maximum degree 2 and trees. The latter result can be thought of as a generalization of that for Vertex Cover Reconfiguration.

On Petrie cycle and Petrie tour partitions of 3- and 4-regular plane graphs

AbstractGiven a plane graph $G=(V,E)$ , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection ${\mathscr P}=\{P_1,\ldots,P_q\}$ of Petrie tours so that each edge of G is in exactly one tour $P_i \in {\mathscr P}$ . A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection ${\mathscr C}=\{C_1,\ldots,C_p\}$ of Petrie cycles so that each vertex of G is in exactly one cycle $C_i \in {\mathscr C}$ . In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph $G=(V,E)$ , a 3-regularization of G is a 3-regular plane graph $G_3$ obtained from G by splitting every vertex $v\in V$ into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.

Approximate and exact results for the harmonious chromatic number

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerned with the computational aspects of finding a minimum harmonious coloring and in the second part we determine the exact value of the harmonious chromatic number for some particular graphs and classes of graphs. More precisely, in the first part we show that finding a minimum harmonious coloring for arbitrary graphs is APX-hard, the natural greedy algorithm is a $\Omega(\sqrt{n})$-approximation, and, moreover, we show a relationship between the vertex cover and the harmonious chromatic number. In the second part we determine the exact value of the harmonious chromatic number for all 3-regular planar graphs of diameter 3, some non-planar regular graphs and cycle-related graphs.

Nice pairs of pentagons in chamfered fullerenes

Fullerenes graphs are 3-connected, 3-regular planar graphs with faces including only pentagons and hexagons. If be a graph with a perfect matching, a subgraph H of G is a nice subgraph if G-V(H) has a perfect matching. In this paper, we show that in every fullerene graph arising from smaller fullerenes via chamfer transformation, each pair of pentagons is a nice subgraph.

Sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate

A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. It is known that every planar graph of girth 6 (equivalently, without 3-, 4-, and 5-cycles) is 2-degenerate. On the other hand, there exist planar graphs without 4 and 5-cycles such as a truncated dodecahedral graph that are not 2-degenerate. Furthermore, a truncated dodecahedral graph also contains none of 6-, 7-, 8-, and 9-cycles. This motivates us to find sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate.In this work, we investigate the degeneracy of planar graphs without 4-, 5-, j-, and k-cycles where j∈{6,7,8,9} and k∈{10,11}. For each j∈{6,9}, we give an example of a 3-regular planar graph without 4-, 5-, j-, 10-, and 11-cycles. In contrast, we prove that every planar graph without 4-, 5-, j-, and k-cycles is 2-degenerate for each j∈{7,8} and k∈{10,11}.

Planar graphs without specific cycles are 2-degenerate

A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, …, and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph is a 3-regular planar graph without 4-, 6-, and 8-cycles. As a related result, we also show that every planar graph without 4-, 6-, 9-, and 10-cycles is 2-degenerate.

Construction of different peculiar structures of 4-regular planar graph with odd regions and even regions and its application

In this article we have proposed the construction of a structure of the 4-regular planar graphs for G(2m+2,4m+4) where m≥2. Based on the proposed structure we have specified two theorems on odd regions and total regions of 4-regular planar graphs. The experimental results and proof of the specified theorems have also been provided. Maximum region covered by odd region is also discussed using the structures of the 4-regular graph. Finally an application is given in region base map coloring and GSM network coloring.

On dispersable book embeddings

In a dispersable book embedding, the vertices of a given graph G must be ordered along a line ℓ, called spine, and the edges of G must be drawn in different half-planes bounded by ℓ, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graph induced by the edges of each page is 1-regular (or equivalently, a matching). The minimum number of pages needed by any dispersable book embedding of G is referred to as the dispersable book thicknessdbt(G) of G. Graph G is called dispersable if dbt(G)=Δ(G) holds (note that Δ(G)≤dbt(G) always holds).Back in 1979, Bernhart and Kainen conjectured that any Δ-regular bipartite graph G is dispersable, i.e., dbt(G)=Δ. In this paper, we employ a counting argument to disprove this conjecture for any fixed value of Δ≥3. Additionally, for the cases Δ=3 and Δ=4 we present concrete counterexamples to the conjecture. In particular, we show that the Gray graph, which is 3-regular and bipartite, has dispersable book thickness four (with a computer-aided proof), while the Folkman graph, which is 4-regular and bipartite, has dispersable book thickness five (with a purely combinatorial proof). On the positive side, we prove that 3-regular bipartite planar graphs are dispersable.

Construction of different peculiar structures of 4-regular planar graph with odd regions and even regions and its application

In this article we have proposed the construction of a structure of the 4-regular planar graphs for G(2m+2,4m+4) where m≥2. Based on the proposed structure we have specified two theorems on odd regions and total regions of 4-regular planar graphs. The experimental results and proof of the specified theorems have also been provided. Maximum region covered by odd region is also discussed using the structures of the 4-regular graph. Finally an application is given in region base map coloring and GSM network coloring.

On the structure of 4-regular planar well-covered graphs

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. It is unknown whether membership in this class of graphs is polynomially decidable. In this paper we focus on the study of well-covered graphs which are 4-regular and planar.

The ropelengths of knots are almost linear in terms of their crossing numbers

For a knot or link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text] and [Formula: see text] be the crossing number of [Formula: see text]. In this paper, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form [Formula: see text]. The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of [Formula: see text] is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order [Formula: see text]. The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).

Construction of different structures of 4-Regular Planar Graph with odd region and even region

Construction of different structures of 4-Regular Planar Graph with odd region and even region

Edge-outer graph embedding and the complexity of the DNA reporter strand problem

In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a DNA reporter strand, that is, a closed walk covering every edge either once or twice, in opposite directions if twice, and passing through each vertex in a particular way. This corresponds to showing that every graph has an edge-outer embedding, that is, an orientable embedding with some face that is incident with every edge. In the motivating application, the objective is such a closed walk of minimum length. Here we give a short algorithmic proof of the original existence result, and also prove that finding a shortest length solution is NP-hard, even for 3-connected cubic (3-regular) planar graphs. Independent of the motivating application, this problem opens a new direction in the study of graph embeddings, and we suggest several problems emerging from it.

Enumeration of labelled 4‐regular planar graphs

We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.

Even Subgraph Expansions for the Flow Polynomial of Planar Graphs with Maximum Degree at Most 4

As projections of links, 4-regular plane graphs are important in combinatorial knot theory. The flow polynomial of 4-regular plane graphs has a close relation with the two-variable Kauffman polynomial of links. F. Jaeger in 1991 provided even subgraph expansions for the flow polynomial of cubic plane graphs. Starting from and based on Jaeger's work, by introducing splitting systems of even subgraphs, we extend Jaeger's results from cubic plane graphs to plane graphs with maximum degree at most 4 including 4-regular plane graphs as special cases. Several consequences are derived and further work is discussed.