Bipartite Planar Research Articles

Exact square coloring of subcubic planar graphs

We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted χ[♯2](G). This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite planar graphs and subcubic K4-minor-free graphs have exact square chromatic number at most 4. We then turn our attention to the class of fullerene graphs, which are cubic planar graphs with face sizes 5 and 6. We characterize fullerene graphs with exact square chromatic number 3. Furthermore, supporting a conjecture of Chen, Hahn, Raspaud and Wang (that all subcubic planar graphs are injectively 5-colorable) we prove that any induced subgraph of a fullerene graph has exact square chromatic number at most 5. This is done by first proving that a minimum counterexample has to be on at most 80 vertices and then computationally verifying the claim for all such graphs.

Slit-Slide-Sew Bijections for Bipartite and Quasibipartite Plane Maps

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward the vertex. The faces of the new map have the same degree as those of the original map, except at the locations of the distinguished corners, where each receives an extra degree: this is the location of the distinguished half-edges. This bijection provides a sampling algorithm for uniform maps with prescribed face degrees and allows to recover Tutte's famous counting formula for bipartite and quasibipartite plane maps.
 In addition, we explain how to decompose the previous bijection into two more elementary ones, which each transfer a degree from one face of the map to another face. In particular, these transfer bijections are simpler to manipulate than the previous one and this point of view simplifies the proofs.

Complexity of Fall Coloring for Restricted Graph Classes

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into $k$ independent dominating sets is NP-complete for every $k \geq 3$. We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451-6458) by showing that it is NP-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every $k \geq 3$, we show that there is some constant $t$ depending only on $k$ such that deciding whether a $k$-regular graph can be partitioned into $t$ independent dominating sets is NP-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

Complexity of planar signed graph homomorphisms to cycles

We study homomorphism problems of signed graphs from a computational point of view. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept when studying signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. The concept of s-homomorphisms has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Nevertheless, such a dichotomy is not known when we restrict the input graph to be planar, not even for non-signed graph homomorphisms.We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square Ct2 of a cycle with t⩾6, or to the circular clique K4t∕(2t−1) with t⩾2, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target.We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree 3 (except for the case of unbalanced 4-cycles, for which we show this for maximum degree 4). We also show that for a given integer g, the problem for signed bipartite planar inputs of girth g is either trivial or NP-complete.

Construction of Barnette graphs whose large subgraphs are non-Hamiltonian

Abstract Barnette’s conjecture states that every three connected cubic bipartite planar graph (CPB3C) is Hamiltonian. In this paper we show the existence of a family of CPB3C Hamiltonian graphs in which large and large subgraphs are non-Hamiltonian.

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.

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3

In this short note, we show that every bipartite planar graph with minimum degree at least 3 has proper orientation number at most 3.

Note on 3-choosability of planar graphs with maximum degree 4

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.

A note on the smallest connected non-traceable cubic bipartite planar graph

A graph is traceable if it contains a Hamiltonian path. We present a connected non-traceable cubic bipartite planar graph with 52 vertices and prove that there are no smaller such graphs.

Complexity and lowers bounds for Power Edge Set Problem

This article is devoted to the study of the complexity of Power Edge Set (PES), a problem occurring when monitoring power networks. We show that PES remains NP-hard in bipartite planar graphs with bounded degree. This result is extended to unit disk graphs and grids with maximum degree three. To the best of our knowledge, this is the most restricted class of graphs known so far on which Power Edge Set is NP-complete. We also show that there is no 2o(n)-time algorithm for Power Edge Set, and there is no 2o(k)nO(1)-time algorithm for its parameterized version, even in bipartite planar subcubic graphs and unit disk graphs, assuming ETH. Finally, we show a polynomial-time algorithm for graphs of bounded treewidth (XP wrt. treewidth), such as series-parallel graphs or outerplanar graphs.

NP-completeness results for partitioning a graph into total dominating sets

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by dt(G). We extend considerably the known hardness results by showing it is ▪-complete to decide whether dt(G)≥3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k≥3, it is ▪-complete to decide whether dt(G)≥k, where G is split or k-regular. In particular, these results complement recent combinatorial results regarding dt(G) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2nnO(1) time, and derive even faster algorithms for special graph classes.

Maximal Planar Subgraphs of Fixed Girth in Random Graphs

In 1991, Bollobás and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$.

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behavior of uniform random maps with a prescribed face‐degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov–Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q‐angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so‐called “discrete snakes” obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.

Clar structures vs Fries structures in hexagonal systems

A hexagonal system H is a 2-connected bipartite plane graph such that all inner faces are hexagons, which is often used to model the structure of a benzenoid hydrocarbon or graphen. A perfect matching of H is a set of disjoint edges which covers all vertices of H. A resonant set S of H is a set of hexagons in which every hexagon is M-alternating for some perfect matching M. The Fries number of H is the size of a maximum resonant set and the Clar number of H is the size of a maximum independent resonant set (i.e. all hexagons are disjoint). A pair of hexagonal systems with the same number of vertices is called a contra-pair if one has a larger Clar number but the other has a larger Fries number. In this paper, we investigates the Fries number and Clar number for hexagonal systems, and show that a catacondensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs.

Oriented, 2-edge-colored, and 2-vertex-colored homomorphisms

Two graph parameters are equivalent if, for every graph class, they are either both bounded or both unbounded. We provide a list of graph parameters equivalent to the acyclic chromatic number, which contains in particular the 2-edge-colored chromatic number. Recently, the CSP dichotomy conjecture has been reduced to the case of 2-edge-colored homomorphism and to the case of 2-vertex-colored homomorphism. Both reductions are rather long and follow the reduction to the case of oriented homomorphism in “Graphs and homomorphisms” by Hell and Nešetřil. We give another proof for the case of 2-vertex-colored homomorphism via a simple reduction from the case of 2-edge-colored homomorphism. Finally, we prove that deciding if the 2-edge-colored chromatic number of a 2-edge-colored graph is at most 4 is NP-complete, even if restricted to 2-connected subcubic bipartite planar graphs with arbitrarily large girth.

Induced forests in bipartite planar graphs

Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.

Complexity of rainbow vertex connectivity problems for restricted graph classes

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems Rainbow Vertex Connectivity and Strong Rainbow Vertex Connectivity, respectively. We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k-regular graphs for k≥3. We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP-complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while Strong Rainbow Vertex Connectivity is tractable for cactus graphs and split graphs.

Large Induced Forests in Graphs

AbstractIn this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least . The second is that every connected triangle‐free graph of order n and size m has an induced forest of order at least . This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound of Alon–Mubayi–Thomas for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least . The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least , where was conjectured to be the best lower bound by Kowalik, Lužar, and Škrekovski.

Rescaled bipartite planar maps converge to the Brownian map

Pour tout entier $n$ strictement positif, on considere une carte planaire aleatoire $\mathcal{M}_{n}$ de loi uniforme sur l’ensemble des cartes biparties enracinees a $n$ aretes. On montre que l’ensemble des sommets de $\mathcal{M}_{n}$ muni de la distance de graphe renormalisee par $(2n)^{-1/4}$ converge en loi au sens de Gromov–Hausdorff vers la carte brownienne. Ce travail complete une serie de resultats de convergence de differents modeles de cartes aleatoires vers la carte brownienne.

Adjacent vertex distinguishing index of bipartite planar graphs

A proper edge coloring of a graph $G$ is called adjacent vertex distinguishing, if any pair of adjacent vertices meet distinct color sets. The adjacent vertex distinguishing index of $G$, denoted by $\chi_a(G)$, is the smallest integer $k$ such that $G$ has an adjacent vertex distinguishing edge $k$-coloring. In this paper, we show that every bipartite planar graph $G$ with maximum degree $\Delta\ge 7$ and without isolated edges has $\chi_{a