Subcubic Graphs Research Articles

Packing [formula omitted]-coloring of some subcubic graphs

For a sequence of non-decreasing positive integers S=(s1,…,sk), a packing S-coloring is a partition of V(G) into sets V1,…,Vk such that for each 1≤i≤k the distance between any two distinct x,y∈Vi is at least si+1. The smallest k such that G has a packing (1,2,…,k)-coloring is called the packing chromatic number of G and is denoted by χp(G). For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The question whether χp(D(G))≤5 for all subcubic graphs G was first asked by Gastineau and Togni and later conjectured by Brešar, Klavžar, Rall and Wash. Gastineau and Togni observed that if one can prove every subcubic graph except the Petersen graph is packing (1,1,2,2)-colorable then the conjecture holds. The maximum average degree, mad(G), is defined to be max{2|E(H)||V(H)|:H⊂G}. In this paper, we prove that subcubic graphs with mad(G)<3011 are packing (1,1,2,2)-colorable. As a corollary, the conjecture of Brešar et al holds for every subcubic graph G with mad(G)<3011.

Decomposing claw-free subcubic graphs and 4-chordal subcubic graphs

Hoffmann–Ostenhof’s conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a 2-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and 4-chordal subcubic graphs.

Integral signed subcubic graphs

A signed graph is called integral if all eigenvalues of its adjacency matrix are integers. It is well-known that there are thirteen connected integral cubic graphs. In this paper, we determine all connected integral signed subcubic graphs, among these signed graphs, there are two classes of integral signed cubic graphs, one contains an infinite family of connected integral signed cubic graphs and another contains 14 sporadic connected integral signed cubic graphs.

A Short Proof of 7-Colorability of $$\frac{3}{3}$$-Power of Subcubic Graphs

For any \(k\in {\mathbb {N}}\), the k-subdivision of graph G is a simple graph \(G^{\frac{1}{k}}\), which is constructed by replacing each edge of G with a path of length k. In (Iradmusa in Discrete Math 310(10–11):1551–1556, 2010), the mth power of the n-subdivision of G has been introduced as a fractional power of G, denoted by \(G^{\frac{m}{n}}\). Wang and Liu (Discrete Math Algorithms Appl 10(3):1850041, 2018) showed that \(\chi (G^{\frac{3}{3}})\le 7\) for any subcubic graph G. In this note, a short proof is given for this theorem by use of incidence chromatic number.

Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width

A semitotal dominating set of a graph G with no isolated vertex is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The minimum size γt2(G) of a semitotal dominating set of G is squeezed between the domination number γ(G) and the total domination number γt(G).Semitotal Dominating Set is the problem of finding, given a graph G, a semitotal dominating set of G of size γt2(G). In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph classes. In particular, we show that it is solvable in polynomial time for the class of graphs of bounded mim-width by a reduction to Total Dominating Set and we provide several approximation lower bounds for subclasses of subcubic graphs. Moreover, we obtain complexity dichotomies in monogenic classes for the decision versions of Semitotal Dominating Set and Total Dominating Set.Finally, we show that it is NP-complete to recognise the graphs such that γt2(G)=γt(G) and those such that γ(G)=γt2(G), even if restricted to be planar and with maximum degree at most 4, and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities.

An Improved Analysis of the Mömke--Svensson Algorithm for Graph-TSP on Subquartic Graphs

Momke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yields a 4/3-approximation guarantee on subcubic graphs as well as a su...

子立方图的严格邻点可区别全染色

子立方图的严格邻点可区别全染色

Two-Distance Vertex-Distinguishing Index of Sparse Subcubic Graphs

The 2-distance vertex-distinguishing index $$\chi '_\mathrm{d2}(G)$$ of a graph G is the minimum number of colors required for a proper edge coloring of G such that any pair of vertices at distance two have distinct sets of colors. It was conjectured that every subcubic graph G has $$\chi '_{\mathrm{d2}}(G)\le 5$$. In this paper, we confirm this conjecture for subcubic graphs with maximum average degree less than $$\frac{8}{3}$$.

On the equality of the induced matching number and the uniquely restricted matching number for subcubic graphs

For a matching M in a graph G, let G(M) be the subgraph of G induced by the vertices of G that are incident with an edge in M. The matching M is induced, if G(M) is 1-regular, and M is uniquely restricted, if M is the unique perfect matching of G(M). The induced matching number νs(G) of G is the largest size of an induced matching in G, and the uniquely restricted matching number νur(G) of G is the largest size of a uniquely restricted matching in G.Golumbic et al. (2001) [4] posed the problem to characterize the graphs G with νs(G)=νur(G). We give a complete characterization of the 2-connected subcubic graphs G of sufficiently large order with νs(G)=νur(G). As a consequence, we are able to show that the subcubic graphs G with νs(G)=νur(G) can be recognized in polynomial time.

2-Coloring number revisited

2-Coloring number is a parameter, which is often used in the literature to bound the game chromatic number and other related parameters. However, this parameter has not been precisely studied before.In this paper we aim to fill this gap. In particular we show that the approximation of the game chromatic number by the 2-coloring number can be very poor for many graphs. Additionally we prove that the 2-coloring number may grow quadratically as a function of the maximum degree of a graph, whereas the game chromatic number is always at most linear.Moreover, we establish the values of the 2-coloring number for several graph classes, such as complete k-partite graphs, cacti and thorny graphs, trees and subcubic graphs. It is shown that in all these cases one may compute the exact value in a polynomial time.

On the minimum energy of regular graphs

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for each element of order n, say G, E(G)≥1.24n, and conjecture that if 6|n, then minimum energy occurs just for each element of this family. We conjecture that there exists N such that for every connected cubic graph G of order n≥N, E(G)≥1.24n.

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.

Acyclic improper choosability of subcubic graphs

A d-improper k-coloring of a graph G is a mapping φ:V(G)→{1,2,…,k} such that for every color i, the subgraph induced by the vertices of color i has maximum degree d. That is, every vertex can be adjacent to at most d vertices with being the same color as itself. Such a d-improper k-coloring is further said to be acyclic if for every pair of distinct colors, say i and j, the induced subgraph by the edges whose endpoints are colored with i and j is a forest. Meanwhile, we say that G is acyclically (k, d)*-colorable.A graph G is called acyclically d-improper L-colorable if for a given list assignment L={L(v)∣v∈V(G)}, there exists an acyclic d-improper coloring φ such that φ(v) ∈ L(v) for each vertex v. If G is acyclically d-improper L-colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V, then we say that G is acyclically d-improper k-choosable, or simply say that G is acyclically (k, d)*-choosable. It is known that every subcubic graph is acyclically (2, 2)*-colorable. But there exists a 3-regular graph that is not necessarily acyclically (2, 2)*-choosable. In this paper, we shall prove that every non-3-regular subcubic graph is acyclically (2, 2)*-choosable.

Uniquely restricted matchings in subcubic graphs

A matching M in a graph G is uniquely restricted if no other matching in G covers the same set of vertices. We conjecture that every connected subcubic graph with m edges and b bridges that is distinct from K3,3 has a uniquely restricted matching of size at least m+b6, and we establish this bound with b replaced by the number of bridges that lie on a path between two vertices of degree at most 2. Moreover, we prove that every connected subcubic graph of order n and girth at least 7 has a uniquely restricted matching of size at least n−13, which partially confirms a conjecture of Fürst and Rautenbach (Graphs Combin. 35 (2019) 353–361).

On Upper Total Domination Versus Upper Domination in Graphs

A total dominating set of a graph G is a dominating set S of G such that the subgraph induced by S contains no isolated vertex, where a dominating set of G is a set of vertices of G such that each vertex in $$V(G){\setminus } S$$ has a neighbor in S. A (total) dominating set S is said to be minimal if $$S{\setminus } \{v\}$$ is not a (total) dominating set for every $$v\in S$$ . The upper total domination number $$\varGamma _t(G)$$ and the upper domination number $$\varGamma (G)$$ are the maximum cardinalities of a minimal total dominating set and a minimal dominating set of G, respectively. For every graph G without isolated vertices, it is known that $$\varGamma _t(G)\le 2\varGamma (G)$$ . The case in which $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$ has been studied in Cyman et al. (Graphs Comb 34:261–276, 2018), which focused on the characterization of the connected cubic graphs and proposed one problem to be solved and two questions to be answered in terms of the value of $$\frac{\varGamma _t(G)}{\varGamma (G)}$$ . In this paper, we solve this problem, i.e., the characterization of the subcubic graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$ , by constructing a class of subcubic graphs, which we call triangle-trees. Moreover, we show that the answers to the two questions are negative by constructing connected cubic graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}$$ and a class of regular non-complete graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$ .

Decomposition of cubic graphs related to Wegner’s conjecture

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red–blue vertex coloring such that the blue subgraph has maximum degree 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. We prove the conjecture for Generalized Petersen graphs.We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.

Packing Chromatic Number of Subdivisions of Cubic Graphs

A packing k-coloring of a graph G is a partition of V(G) into sets $$V_1,\ldots ,V_k$$ such that for each $$1\le i\le k$$ the distance between any two distinct $$x,y\in V_i$$ is at least $$i+1$$ . The packing chromatic number, $$\chi _p(G)$$ , of a graph G is the minimum k such that G has a packing k-coloring. For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The questions on the value of the maximum of $$\chi _p(G)$$ and of $$\chi _p(D(G))$$ over the class of subcubic graphs G appear in several papers. Gastineau and Togni asked whether $$\chi _p(D(G))\le 5$$ for any subcubic G, and later Bresar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $$\chi _p(G)$$ is not bounded in the class of subcubic graphs G. In contrast, in this paper we show that $$\chi _p(D(G))$$ is bounded in this class, and does not exceed 8.

The average degree of a subcubic edge‐chromatic critical graph

AbstractA graph is here called 3‐ critical if , and for every edge of . The 3‐critical graphs include (the Petersen graph with a vertex deleted), and subcubic graphs that are Hajós joins of copies of . Building on a recent paper of Cranston and Rabern, it is proved here that if is 3‐critical and not nor a Hajós join of two copies of , then has average degree at least ; this bound is sharp, as it is the average degree of a Hajós join of three copies of .

Global edge alliances in graphs

In the paper we introduce and study a new problem of finding a minimum global edge alliance in a graph which is related to the global defensive alliance (Haynes et al., 2013; Hedetniemi, 2004) and the global defensive set (Lewoń et al., 2016). We proved the NP-completeness of the global edge alliance problem for subcubic graphs and we constructed polynomial time algorithms for trees. We found the exact values of the size of the minimum global edge alliance for certain classes: paths, cycles, wheels, complete k-partite graphs and complete k-ary trees. Moreover, we proved some lower bounds for arbitrary graphs.