3-connected Planar Graphs Research Articles

Cyclic coloring of plane graphs with maximum face size 16 and 17

Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size Δ⋆ can be colored using at most Δ⋆+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for Δ⋆=3, Δ⋆=4 and Δ⋆≥18. We prove the conjecture for Δ⋆=16 and Δ⋆=17.

Coloring near-quadrangulations of the cylinder and the torus

Let G be a simple connected plane graph and let C1 and C2 be cycles in G bounding distinct faces f1 and f2. For a positive integer ℓ, let r(ℓ) denote the number of integers n such that −ℓ≤n≤ℓ, n is divisible by 3, and n has the same parity as ℓ; in particular, r(4)=1. Let rf1,f2(G)=∏fr(|f|), where the product is over the faces f of G distinct from f1 and f2, and let q(G)=1+∑f:|f|≠4|f|, where the sum is over all faces f of G (of length other than four). We give an algorithm with time complexity O(rf1,f2(G)q(G)|G|) which, given a 3-coloring ψ of C1∪C2, either finds an extension of ψ to a 3-coloring of G, or correctly decides no such extension exists.The algorithm is based on a min–max theorem for a variant of integer 2-commodity flows, and consequently in the negative case produces an obstruction to the existence of the extension. As a corollary, we show that every triangle-free graph drawn in the torus with edge-width at least 21 is 3-colorable.

On the cyclic coloring conjecture

A cyclic coloring of a plane graph G is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph G is its cyclic chromatic number χc(G). Let Δ∗(G) be the maximum face degree of a graph G.In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected plane graph G has χc(G)≤⌊32Δ∗(G)⌋, it is enough to do it for subdivisions of simple 3-connected plane graphs.We have discovered four new different upper bounds on χc(G) for graphs G from this restricted family; three bounds of them are tight. As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple 3-connected plane quadrangulations, and simple 3-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple 3-connected plane graphs of maximum degree at least 10, and for subdivisions of simple 3-connected plane graphs having the maximum face degree large enough in comparison with the number of vertices of their longest paths consisting only of vertices of degree two.

A counterexample to prism-hamiltonicity of 3-connected planar graphs

The prism over a graph G is the Cartesian product of G with the complete graph K2. A graph G is hamiltonian if there exists a spanning cycle in G, and G is prism-hamiltonian if the prism over G is hamiltonian.Rosenfeld and Barnette (1973) [15] conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.

Gaps in the cycle spectrum of 3-connected cubic planar graphs

We prove that, for every natural number k, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in [k,2k+9]. We also show that this bound is close to being optimal by constructing, for every even k≥4, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in [k,2k+1].

Unique chords of unique cycles in 3-connected planar graphs

Edges that are the unique chords of at least one cycle have been studied by a variety of authors over the past dozen years. This paper begins the study of those graphs in which each edge is the unique chord of exactly one cycle. The [Formula: see text]-connected planar graphs that enjoy this restriction are characterized by two infinite sequences (the dipyramid and trapezohedron [Formula: see text]-polytopes) together with three special graphs (the [Formula: see text]-antiprism, and the [Formula: see text]- and [Formula: see text]-prism graphs).

Tri-partitions and Bases of an Ordered Complex

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.

The Braess' paradox for pendent twins

Kemeny's constant κ(G) of a connected undirected graph G can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with G. In certain cases, inserting a new edge into G has the counter-intuitive effect of increasing the value of κ(G). In the current work we identify a large class of graphs exhibiting this “paradoxical” behavior – namely, those graphs having a pair of twin pendent vertices. We also investigate this phenomenon in the context of random graphs, showing that it occurs for almost all connected planar graphs. To establish these results, we make use of a connection between Kemeny's constant and the resistance distance of graphs.

Domination number of annulus triangulations

An {\em annulus triangulation} $G$ is a 2-connected plane graph with two disjoint faces $f_1$ and $f_2$ such that every face other than $f_1$ and $f_2$ are triangular, and that every vertex of $G$ is contained in the boundary cycle of $f_1$ or $f_2$. In this paper, we prove that every annulus triangulation $G$ with $t$ vertices of degree 2 has a dominating set with cardinality at most $\lfloor \frac{|V(G)|+t+1}{4} \rfloor$ if $G$ is not isomorphic to the octahedron. In particular, this bound is best possible.

Facial rainbow edge-coloring of simple 3-connected plane graphs

A facial rainbow edge-coloring of a plane graph \(G\) is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of \(G\). The minimum number of colors used in such a coloring is denoted by \(\text{erb}(G)\). Trivially, \(\text{erb}(G) \geq \text{L}(G)+1\) holds for every plane graph without cut-vertices, where \(\text{L}(G)\) denotes the length of a longest facial path in \(G\). Jendroľ in 2018 proved that every simple \(3\)-connected plane graph admits a facial rainbow edge-coloring with at most \(\text{L}(G)+2\) colors, moreover, this bound is tight for \(\text{L}(G)=3\). He also proved that \(\text{erb}(G) = \text{L}(G)+1\) for \(\text{L}(G)\not\in\{3,4,5\}\). He posed the following conjecture: There is a simple \(3\)-connected plane graph \(G\) with \(\text{L}(G)=4\) and \(\text{erb}(G)=\text{L}(G)+2\). In this note we answer the conjecture in the affirmative.

Improved bounds for some facially constrained colorings

A facial-parity edge-coloring of a $2$-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a $2$-connected plane graph is a facially-proper vertex-coloring in which every face is incident with zero or an odd number of vertices of each color. Czap and Jendro\v{l} (in Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017), 2691--2703), conjectured that $10$ colors suffice in both colorings. We present an infinite family of counterexamples to both conjectures. A facial $(P_{k}, P_{\ell})$-WORM coloring of a plane graph $G$ is a coloring of the vertices such that $G$ contains no rainbow facial $k$-path and no monochromatic facial $\ell$-path. Czap, Jendro\v{l} and Valiska (in WORM colorings of planar graphs, Discuss. Math. Graph Theory 37 (2017), 353--368), proved that for any integer $n\ge 12$ there exists a connected plane graph on $n$ vertices, with maximum degree at least $6$, having no facial $(P_{3},P_{3})$-WORM coloring. They also asked if there exists a graph with maximum degree $4$ having the same property. We prove that for any integer $n\ge 18$, there exists a connected plane graph, with maximum degree $4$, with no facial $(P_{3},P_{3})$-WORM coloring.

On the Circumference of Essentially 4-connected Planar Graphs

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.

The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3

We prove that the Weisfeiler--Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best-known upper bounds for the dimension and number of variables were 14 and 15, respectively. First, we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of every arc-colored 3-connected graph belonging to this class. Then, we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two appropriately chosen vertices of a colored 3-connected planar graph followed by the one-dimensional WL-algorithm produces the discrete vertex partition. This implies that the three-dimensional WL-algorithm determines the orbits of arc-colored 3-connected planar graphs. As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.

Remarks on Barnette\u2019s conjecture

Let P be a cubic 3-connected bipartite plane graph having a 2-factor which consists only of facial 4-cycles, and suppose that P^{*} is the dual graph. We show that P has at least 3^{leftlceil frac{2|P^{*}|}{varDelta ^{2}{(P^{*})}}rightrceil } different Hamilton cycles.

The total face irregularity strength of some plane graphs

A face irregular total k-labeling λ:V∪E→{1,2,…,k} of a 2-connected plane graph G is a labeling of vertices and edges such that their face-weights are pairwise distinct. The weight of a face f under a labeling λ is the sum of the labels of all vertices and edges surrounding f. The minimum value k for which G has a face irregular total k-labeling is called the total face irregularity strength of G, denoted by tfs(G). The lower bound of tfs(G) is provided along with the exact value of two certain plane graphs. Improving the results, this paper deals with the total face irregularity strength of the disjoint union of multiple copies of a plane graph G. We estimate the bounds of tfs(G) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.

The one-cop-moves game on planar graphs

Cops and Robbers is a vertex-pursuit game played on graphs. In this game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph’s edges with perfect information about each other’s positions. If a cop eventually occupies the same vertex as the robber, then the cops win; the robber wins if she can indefinitely evade capture. Aigner and Fromme established that in every connected planar graph, three cops are sufficient to capture a single robber. In this paper, we consider a recently studied variant of the cops and robbers game, alternately called the one-active-cop game, one-cop-moves game or the lazy cops and robbers game, where at most one cop can move during any round. We show that Aigner and Fromme’s result does not generalize to this game variant by constructing a connected planar graph on which a robber can indefinitely evade three cops in the one-cop-moves game.

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.

Strong edge-coloring for planar graphs with large girth

A strong edge-coloring of a graph G=(V,E) is a partition of its edge set E into induced matchings. Let G be a connected planar graph with girth k≥26 and maximum degree Δ. We show that either G is isomorphic to a subgraph of a very special Δ-regular graph with girth k, or G has a strong edge-coloring using at most 2Δ+⌈12(Δ−2)k⌉ colors.

Coloring of fullerenes

A (k,6)-fullerene graph is a planar 3-connected cubic graph whose faces are k-gons and hexagons. The aim of this paper is to compute the equitable chromatic, b-chromatic and list chromatic numbers of some fullerenes.

Connectivity of Minimum Non-5-injectively Colorable Planar Cubic Graphs

Suppose that G is a planar cubic graph with $$\chi _i(G)>5$$. We show that if $$\chi _i(H)<\chi _i(G)$$ for each planar cubic graph H of order less than G, then G is either a 3-connected simple planar cubic graph, or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings. This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.