Subcubic Graphs Research Articles

Between proper and strong edge‐colorings of subcubic graphs

AbstractIn a proper edge‐coloring the edges of every color form a matching. A matching is induced if the end‐vertices of its edges induce a matching. A strong edge‐coloring is an edge‐coloring in which the edges of every color form an induced matching. We consider intermediate types of edge‐colorings, where edges of some colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper of Gastineau and Togni on ‐packing edge‐colorings asserting that, for subcubic graphs, by allowing three additional induced matchings, one is able to save one matching color. We prove that every graph with maximum degree 3 can be decomposed into one matching and at most eight induced matchings, and two matchings and at most five induced matchings. We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the above‐mentioned conjecture for class I graphs.

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.

Strong edge colorings of graphs and the covers of Kneser graphs

AbstractA proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a ‐regular graph at least colors are needed. We show that a ‐regular graph admits a strong edge coloring with colors if and only if it covers the Kneser graph . In particular, a cubic graph is strongly 5‐edge‐colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. is false.

A New Proof for a Result on the Inclusion Chromatic Index of Subcubic Graphs

Let G be a graph with a minimum degree δ of at least two. The inclusion chromatic index of G, denoted by χ⊂′(G), is the minimum number of colors needed to properly color the edges of G so that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. We prove that every connected subcubic graph G with δ(G)≥2 either has an inclusion chromatic index of at most six, or G is isomorphic to K^2,3, where its inclusion chromatic index is seven.

On Computing the Hamiltonian Index of Graphs ⋆

The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest $r$ such that $L^{r}(G)$ has a Hamiltonian cycle. Checking if $h(G) = k$ is NP-hard for any fixed integer $k \geq 0$ even for subcubic graphs $G$. We study the parameterized complexity of this problem with the parameter treewidth, $tw(G)$, and show that we can find $h(G)$ in time $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ where $\omega$ is the matrix multiplication exponent and the $O*$ notation hides polynomial factors in input size. The NP-hard Eulerian Steiner Subgraph problem takes as input a graph $G$ and a specified subset $K$ of terminal vertices of $G$ and asks if $G$ has an Eulerian (that is: connected, and with all vertices of even degree.) subgraph $H$ containing all the terminals. A second result (and a key ingredient of our algorithm for finding $h(G)$) in this work is an algorithm which solves Eulerian Steiner Subgraph in $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ time.

The neighbour sum distinguishing relaxed edge colouring

A k-edge colouring (not necessarily proper) of a graph with colours in {1,2,…,k} is neighbour sum distinguishing if, for any two adjacent vertices, the sums of the colours of the edges incident with each of them are distinct. The smallest value of k such that such a colouring of G exists is denoted by χ∑e(G). When we add the additional restriction that the k-edge colouring must be proper, then the smallest value of k such that such a colouring exists is denoted by χ∑′(G). Such colourings are studied on a connected graph on at least 3 vertices. There are two famous conjectures on these edge colourings: the 1-2-3 Conjecture states that χ∑e(G)≤3 for any graph G; and the other states that χ∑′(G)≤Δ(G)+2 for any graph G≠C5. In this paper, we generalize these two versions of neighbour sum distinguishing edge colourings by introducing the edge colouring in which each monochromatic set of edges induces a subgraph with maximum degree at most d. We call such an edge colouring that distinguishes adjacent vertices a neighbour sum distinguishing d-relaxed k-edge colouring. We denote by χ∑′d(G) the smallest value of k such that such a colouring of G exists. We study families of graphs for which χ∑′ is known. We show that the number of required colours decreases when the proper condition is relaxed. In particular, we prove that χ∑′2(G)≤4 for every subcubic graph. For complete graphs, we show that χ∑′d(Kn)≤4 if d∈{⌈n−12⌉,…,n−1} and we also determine the exact value of χ∑′2(Kn). Finally, we determine the value of χ∑′d(T) for any tree T.

Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

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.

Cops and Robber on some families of oriented graphs

Cops and Robber is a two-player pursuit-evasion game in which a set of cops, controlled by Player 1, try to capture the robber, controlled by Player 2. We study three models of this game on oriented graphs which differ based on the kind of moves the players can make. These models are (i) the normal cop model, where both cops and robber can only move along the direction of the arcs; (ii) the strong cop model, where the cops can move along or against the direction of the arcs while the robber can only move along them; and (iii) the weak cop model, where the robber can move along or against the direction of the arcs while the cops can only move along them. The cop number is the minimum number of cops required to capture the robber in a graph, and a graph is cop-win if its cop number is 1.We begin our study by comparing the cop number in these three models for oriented graphs. For the normal cop model, we show that there exist strongly connected oriented graphs having high girth, high minimum degree, and high cop number. We also characterize the cop-win graphs in various graph classes like transitive-triangle-free, outerplanar, and subcubic graphs. For the strong cop model, we construct graphs with unbounded cop number, and also study the cop number of grids, outerplanar, and planar graphs. For the weak cop model, we characterize the cop-win graphs.

Decomposing subcubic graphs into claws, paths or triangles

AbstractLet be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph into graphs taken from any nonempty . The problem is known to be ‐complete for any possible choice of in general graphs. In this paper, we assume that the input graph is subcubic (i.e., all its vertices have degree at most 3), and study the computational complexity of the problem of partitioning its edge set for any choice of . We identify all polynomial and ‐complete problems in that setting.

Complexity and inapproximability results for balanced connected subgraph problem

This work is devoted to the study of the Balanced Connected Subgraph Problem (BCS) from a complexity, inapproximability and approximation point of view. The input is a graph G=(V,E), with each vertex having been colored, “red” or “blue”; the goal is to find a maximum connected subgraph G′=(V′,E′) from G that is color-balanced (having exactly |V′|/2 red vertices and |V′|/2 blue vertices). This problem is known to be NP-complete in general but polynomial in paths and trees. We propose a polynomial-time algorithm for block graph. We propose some complexity results for bounded-degree or bounded-diameter graphs, and also for bipartite graphs. We also propose inapproximability results for some graph classes, including chordal, planar, or subcubic graphs.

List injective edge-coloring of subcubic graphs

An edge-coloring of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. In this paper, we consider the list version of the injective edge-coloring and prove that the list-injective chromatic index of every subcubic graph is at most 7, which generalizes the recent result of Kostochka et al. (2021).

On the tree-width of even-hole-free graphs

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.We prove that for every graph G, if G excludes a fixed graph H as a minor, then G either has small tree-width, or G contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour’s excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graphs have bounded tree-width (Silva et al., 2010).We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.

On the in–out–proper orientations of graphs

An orientation of a graph G is in–out–proper if any two adjacent vertices have different in–out-degrees, where the in–out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The in–out–proper orientation number of a graph G, denoted by χ↔(G), is minD∈Γmaxv∈V(G)|dD±(v)|, where Γ is the set of in–out–proper orientations of G and dD±(v) is the in–out-degree of the vertex v in the orientation D. Borowiecki et al. proved that the in–out–proper orientation number is well-defined for any graph G (Borowiecki et al., 2012). So we have χ↔(G)≤Δ(G), where Δ(G) is the maximum degree of vertices in G. We conjecture that there exists a constant number c such that for every planar graph G, we have χ↔(G)≤c. Toward this speculation, we show that for every tree T we have χ↔(T)≤3 and this bound is sharp. Next, we study the in–out–proper orientation number of subcubic graphs. By using the properties of totally unimodular matrices we show that there is a polynomial time algorithm to determine whether χ↔(G)≤2, for a given graph G with maximum degree three. On the other hand, we show that it is NP-complete to decide whether χ↔(G)≤1 for a given bipartite graph G with maximum degree three. Finally, we study the in–out–proper orientation number of regular graphs.

Packing [formula omitted]-coloring of subcubic outerplanar graphs

For 1≤s1≤s2≤…≤sk and a graph G, a packing(s1,s2,…,sk)-coloring of G is a partition of V(G) into sets V1,V2,…,Vk such that, for each 1≤i≤k the distance between any two distinct x,y∈Vi is at least si+1. The packing chromatic number, χp(G), of a graph G is the smallest k such that G has a packing (1,2,…,k)-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs G with arbitrarily large χp(G). Recently, there was a series of papers on packing (s1,s2,…,sk)-colorings of subcubic graphs in various classes. We show that every 2-connected subcubic outerplanar graph has a packing (1,1,2)-coloring and every subcubic outerplanar graph is packing (1,1,2,4)-colorable. Our results are sharp in the sense that there are 2-connected subcubic outerplanar graphs that are not packing (1,1,3)-colorable and there are subcubic outerplanar graphs that are not packing (1,1,2,5)-colorable. We also show subcubic outerplanar graphs that are not packing (1,2,2,4)-colorable and not packing (1,1,3,4)-colorable.

Inclusion total chromatic number

Let G=(V,E) be a graph and c:(V∪E)→C be a proper total colouring of G, where C is a set of colours. We call c inclusion-free if for each vertex, the set of colours appearing on the vertex and the incident edges is not a subset of the respective sets of its neighbours. With a probabilistic argument we show that the minimum number of colours for inclusion-free total colouring, denoted by χ⊂″(G), is bounded from above by Δ+150log⁡Δ for any graph G with large enough maximum degree Δ. Then we prove that χ⊂″(G)≤6 for any subcubic graph G, which meets the bound for adjacent vertex distinguishing total colouring.

A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges

We consider three variants of the problem of finding a maximum weight restricted 2-matching in a subcubic graph G. (A 2-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on the variant a restricted 2-matching means a 2-matching that is either triangle-free or square-free or both triangle- and square-free. Since computing a maximum weight square-free 2-matching in a subcubic graph is NP-hard, in the second and third variant we additionally assume that the edge-weights are vertex-induced on each square. While there exist polynomial time algorithms for the first two types of 2-matchings, they are quite complicated or use advanced methodology. For each of the three problems we present a simple reduction to the computation of a maximum weight b-matching. The reduction is conducted with the aid of half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints. For a subset of triangles and/or squares of G, we replace each edge of such a triangle/square with two half-edges. Two half-edges of one edge e of weight w(e) may get different weights, not necessarily equal to 12w(e). In the metric setting when the edge weights satisfy the triangle inequality, this has a geometric interpretation connected to how an incircle partitions the edges of a triangle. Our algorithms are additionally faster than those known before. The running time of each of them is O(n2log⁡n), where n denotes the number of vertices in the graph.

Exponential independence in subcubic graphs

A set S of vertices of a graph G is exponentially independent if, for every vertex u in S,∑v∈S∖{u}(12)dist(G,S)(u,v)−1<1, where dist(G,S)(u,v) is the distance between u and v in the graph G−(S∖{u,v}). The exponential independence numberαe(G) of G is the maximum order of an exponentially independent set in G. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order n have exponentially independent sets of order Ω(n/log2⁡(n)), that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order n have exponentially independent sets of order (n+3)/4.

Independent domination in subcubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that if G is a connected cubic graph of order n, then $$i(G) \le \frac{3}{8}n$$ , except if G is the complete bipartite graph $$K_{3,3}$$ or the 5-prism $$C_5 \, \Box \, K_2$$ . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. In this paper, we provide a new family of connected cubic graphs G of order n such that $$i(G) = \frac{3}{8}n$$ . We also show that if G is a subcubic graph of order n with no isolated vertex, then $$i(G) \le \frac{1}{2}n$$ , and we characterize the graphs achieving equality in this bound.