Subcubic Planar Graphs Research Articles

The square of every subcubic planar graph of girth at least 6 is 7-choosable

The square of a graph G, denoted G2, has the same vertex set as G and has an edge between two vertices if the distance between them in G is at most 2. Thomassen [16] and Hartke, Jahanbekam and Thomas [8] proved that χ(G2)≤7 if G is a subcubic planar graph. A natural question is whether χℓ(G2)≤7 or not if G is a subcubic planar graph. It was showed in [5] that χℓ(G2)≤7 if G is a subcubic planar graph of girth at least 7. We prove that χℓ(G2)≤7 if G is a subcubic planar graph of girth at least 6.

A note on median eigenvalues of subcubic graphs

Let G be a simple graph of order n whose adjacency eigenvalues are λ1≥⋯≥λn. The HL-index of G is defined to be R(G)=max{|λh|,|λl|} with h=n+12 and l=n+12. Mohar conjectured that R(G)≤1 for every planar subcubic graph G. In this note, we prove that Mohar’s Conjecture holds for every K4-minor-free subcubic graph. In addition, R(G)≤1 for every subcubic graph G which contains a subgraph K2,3.

On the Tractability of Covering a Graph with 2-Clubs

Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science, for example when the cohesive subgraph model considered is a clique. In this paper, we consider as a model of cohesive subgraph the 2-clubs, which are induced subgraphs of diameter at most 2. We prove new complexity results on the mathsf {Min~2text {-}Club~Cover} problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs. First, we answer an open question on the decision version of mathsf {Min~2text {-}Club~Cover} that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of mathsf {Min~2text {-}Club~Cover} on some graph classes. We prove that mathsf {Min~2text {-}Club~Cover} remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution, and fixed-parameter tractable on graphs having bounded treewidth.

Subcubic planar graphs of girth 7 are class I

We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974.

Positive planar satisfiability problems under 3-connectivity constraints

A 3-SAT problem is called positive and planar if all the literals are positive and the clause-variable incidence graph (i.e., SAT graph) is planar. The NAE 3-SAT and 1-in-3-SAT are two variants of 3-SAT that remain NP-complete even when they are positive. The positive 1-in-3-SAT problem remains NP-complete under planarity constraint, but planar NAE 3-SAT is solvable in O(n1.5log⁡n) time, where n is the number of vertices. In this paper we prove that a positive planar NAE 3-SAT is always satisfiable when the underlying SAT graph is 3-connected, and a satisfiable assignment can be obtained in linear time. We also show that without 3-connectivity constraint, existence of a linear-time algorithm for positive planar NAE 3-SAT problem is unlikely as it would imply a linear-time algorithm for finding a spanning 2-matching in a planar subcubic graph. We then prove that positive planar 1-in-3-SAT remains NP-complete under the 3-connectivity constraint, even when each variable appears in at most 4 clauses. However, we show that the 3-connected planar 1-in-3-SAT is always satisfiable when each variable appears in an even number of clauses.

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

Jones' Conjecture in Subcubic Graphs

We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to obtain a forest.

Complexity and algorithms for injective edge-coloring in graphs

An injective k-edge-coloring of a graph G is an assignment of colors, i.e. integers in {1,…,k}, to the edges of G such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a k-coloring exists is called Injectivek-Edge-Coloring. We show that Injective 3-Edge-Coloring is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth 6. Injective 4-Edge-Coloring remains NP-complete for cubic graphs. For any k≥45, we show that Injectivek-Edge-Coloring remains NP-complete even for graphs of maximum degree at most 53k. In contrast with these negative results, we show that Injectivek-Edge-Coloring is linear-time solvable on graphs of bounded treewidth. Moreover, we show that all planar bipartite subcubic graphs of girth at least 16 are injectively 3-edge-colorable. In addition, any graph of maximum degree at most k/2 is injectively k-edge-colorable.

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.

Reducing the domination number of graphs via edge contractions and vertex deletions

In this work, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k>0? We show that for k=1 (resp. k=2), the problem is NP-hard (resp. coNP-hard). We further prove that for k=1, the problem is W[1]-hard parameterized by domination number plus the mim-width of the input graph, and that it remains NP-hard when restricted to chordal {P6,P4+P2}-free graphs, bipartite graphs and {C3,…,Cℓ}-free graphs for any ℓ≥3. We also show that for k=1, the problem is coNP-hard on subcubic claw-free graphs, subcubic planar graphs and on 2P3-free graphs. On the positive side, we show that for any k≥1, the problem is polynomial-time solvable on (P5+pK1)-free graphs for any p≥0 and that it can be solved in FPT-time and XP-time when parameterized by treewidth and mim-width, respectively. Finally, we start the study of the problem of reducing the domination number of a graph via vertex deletions and edge additions and, in this case, present a complexity dichotomy on H-free graphs.

Distance-two colourings of Barnette graphs

Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. We examine these classes from the point of view of distance-two colourings. A distance-two r-colouring of a graph G is an assignment of r colours to the vertices of G so that any two vertices at distance at most two have different colours. A cubic graph obviously needs at least four colours, and the distance-two four-colouring problem for cubic planar graphs is known to be NP-complete. We prove the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we show that the problem is polynomial for cubic plane graphs with face sizes 3,4,5, or 6, which we call type-two Barnette graphs, because of their relation to Barnette’s second conjecture. In fact, the colourable instances that are bipartite can be fully described, and those that are non-bipartite can be characterized by their face sizes. We have similar results for quartic plane graphs: the analogue of type-two Barnette graphs are graphs with face sizes 3 or 4. For this class, the corresponding distance-two five-colouring problem is also polynomial; in fact, we can again fully describe all colourable instances — there are exactly two such graphs. It has recently been proved that every planar subcubic graph can be distance-two coloured with at most seven colours, and conjectured that six colours suffice if the planar subcubic graph can be drawn without faces of size five. We consider a weaker version of the conjecture, stating that six colours suffice for a bipartite cubic planar graph, i.e., a cubic plane graph with all faces even. We prove this conjecture in the case when all faces have sizes divisible by four, and in another more general case, when the faces are coloured in three colours, of which two colours consist of faces with sizes divisible by four.

Large independent sets in triangle-free cubic graphs: beyond planarity

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

Extremal $H$-Free Planar Graphs

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

Algorithmic expedients for the S-labeling problem

Graph labeling problems have been widely studied in the last decades and have a vast area of application. In this work, we study the recently introduced S-labeling problem, in which the nodes get labeled using labels from 1 to |V| and for each edge the contribution to the objective function, called S-labeling number of the graph, is the minimum label of its end-nodes. The goal is to find a labeling with minimum value. The problem is NP-hard for planar subcubic graphs, although for many other graph classes the complexity status is still unknown.In this paper, we present different algorithmic approaches for tackling this problem: We develop an exact solution framework based on Mixed-Integer Programming (MIP) which is enhanced with valid inequalities, starting and primal heuristics and specialized branching rules. We show that our MIP formulation has no integrality gap for paths, cycles and perfect n-ary trees, and, to the best of our knowledge, we give the first polynomial-time algorithm for the problem on n-ary trees as well as a closed formula for the S-labeling number for such trees. Moreover, we also present a Lagrangian heuristic and a constraint programming approach. A computational study is carried out in order to (i) investigate if there may be other special graph classes, where our MIP formulation has no integrality gap, and (ii) assess the effectiveness of the proposed solution approaches for solving the problem on a dataset consisting of general graphs.

Parsimonious edge-coloring on surfaces

We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs.

Strong list-chromatic index of subcubic graphs

A strong k-edge-coloring of a graph G is an edge-coloring with k colors in which every color class is an induced matching. The strong chromatic index of G, denoted by χs′(G), is the minimum k for which G has a strong k-edge-coloring. In 1985, Erdős and Nešetřil conjectured that χs′(G)≤54Δ(G)2, where Δ(G) is the maximum degree of G. When G is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.

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.

Drawing subcubic planar graphs with four slopes and optimal angular resolution

A subcubic planar graph is a planar graph whose vertices have degree at most 3. We show that the subcubic planar graphs with at least five vertices have planar slope number at most 4, which is worst-case optimal. This answers an open question by Jelínek et al. [10]. Furthermore, we prove that the subcubic planar graphs with at least five vertices have angular resolution π/4, which solves an open problem by Kant [11] and by Formann et al. [8].

Injective choosability of subcubic planar graphs with girth 6

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if for any list assignment L, where |L(v)|≥k for all v∈V(G), G has an injective L-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens the result of Lužar, Škrekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.

Polynomial-time solvability of the independent set problem in a certain class of subcubic planar graphs

The independent set problem for a given simple graph consists in computing the size of a largest subset of its pairwise nonadjacent vertices. In this article, we prove the polynomial solvability of the problem for the subcubic planar graphs with no induced tree obtained by identifying the ends of three paths of lengths 3, 3, and 2 respectively.