A greedy algorithm for the minimum 2-connected m-fold dominating set problem

To save energy and alleviate interference in a wireless sensor network, connected dominating set (CDS) has been proposed as the virtual backbone. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a \(k\)-connected \(m\)-fold dominating set \(((k,m)\)-CDS for short): a subset of nodes \(C\subseteq V(G)\) is a \((k,m)\)-CDS of \(G\) if every node in \(V(G)\setminus C\) is adjacent with at least \(m\) nodes in \(C\) and the subgraph of \(G\) induced by \(C\) is \(k\)-connected.In this paper, we present an approximation algorithm for the minimum \((2,m)\)-CDS problem with \(m\ge 2\). Based on a \((1,m)\)-CDS, the algorithm greedily merges blocks until the connectivity is raised to two. The most difficult problem in the analysis is that the potential function used in the greedy algorithm is not submodular. By proving that an optimal solution has a specific decomposition, we managed to prove that the approximation ratio is \(\alpha +2(1+\ln \alpha )\), where \(\alpha \) is the approximation ratio for the minimum \((1,m)\)-CDS problem. This improves on previous approximation ratios for the minimum \((2,m)\)-CDS problem, both in general graphs and in unit disk graphs.     

Upper bounds for the total rainbow connection of graphs

A two-agent scheduling problem on parallel machines is considered. Our objective is to minimize the makespan for agent A, subject to an upper bound on the makespan for agent B. When the number of machines, denoted by \(m\), is chosen arbitrarily, we provide an \(O(n)\) algorithm with performance ratio \(2-\frac{1}{m}\), i.e., the makespan for agent A given by the algorithm is no more than \(2-\frac{1}{m}\) times the optimal value, while the makespan for agent B is no more than \(2-\frac{1}{m}\) times the threshold value. This ratio is proved to be tight. Moreover, when \(m=2\), we present an \(O(nlogn)\) algorithm with performance ratio \(\frac{1+\sqrt{17}}{4}\approx 1.28\) which is smaller than \(\frac{3}{2}\). The ratio is weakly tight.     

On judicious partitions of graphs

Let \(k, m\) be positive integers, let \(G\) be a graph with \(m\) edges, and let \(h(m)=\sqrt{2m+\frac{1}{4}}-\frac{1}{2}\). Bollobás and Scott asked whether \(G\) admits a \(k\)-partition \(V_{1}, V_{2}, \ldots , V_{k}\) such that \(\max _{1\le i\le k} \{e(V_{i})\}\le \frac{m}{k^2}+\frac{k-1}{2k^2}h(m)\) and \(e(V_1, \ldots , V_k)\ge {k-1\over k} m +{k-1\over 2k}h(m) -\frac{(k-2)^{2}}{8k}\). In this paper, we present a positive answer to this problem on the graphs with large number of edges and small number of vertices with degrees being multiples of \(k\). Particularly, if \(d\) is not a multiple of \(k\) and \(G\) is \(d\)-regular with \(m\ge {9\over 128}k^4(k-2)^2\), then \(G\) admits a \(k\)-partition as desired. We also improve an earlier result by showing that \(G\) admits a partition \(V_{1}, V_{2}, \ldots , V_{k}\) such that \(e(V_{1},V_{2},\ldots ,V_{k})\ge \frac{k-1}{k}m+\frac{k-1}{2k}h(m)-\frac{(k-2)^{2}}{2(k-1)}\) and \(\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)\).     

Improved upper bound for the degenerate and star chromatic numbers of graphs

Let \(G=G(V,E)\) be a graph. A proper coloring of G is a function \(f:V\rightarrow N\) such that \(f(x)\ne f(y)\) for every edge \(xy\in E\). A proper coloring of a graph G such that for every \(k\ge 1\), the union of any k color classes induces a \((k-1)\)-degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored \(P_{4}\) is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as \(\chi _{sd}(G)\). In this paper, we employ entropy compression method to obtain a new upper bound \(\chi _{sd}(G)\le \lceil \frac{19}{6}\Delta ^{\frac{3}{2}}+5\Delta \rceil \) for general graph G.     

An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

Approximation for maximizing monotone non-decreasing set functions with a greedy method

We study the problem of maximizing a monotone non-decreasing function \(f\) subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if \(f\) is submodular, the greedy algorithm will find a solution with value at least \(\frac{1}{2}\) of the optimal value under a general matroid constraint and at least \(1-\frac{1}{e}\) of the optimal value under a uniform matroid \((\mathcal {M} = (X,\mathcal {I})\), \(\mathcal {I} = \{ S \subseteq X: |S| \le k\}\)) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least \(\frac{1}{1+\mu }\) of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where \(\mu = \alpha \), if \(0 \le \alpha \le 1\); \(\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}\) if \(\alpha > 1\); here \(\alpha \) is a constant representing the “elemental curvature” of \(f\), and \(K\) is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a \(1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k\) approximation under a uniform matroid constraint. Under this unified \(\alpha \)-classification, submodular functions arise as the special case \(0 \le \alpha \le 1\).     

Online covering salesman problem

Given a graph \(G=(V,E,D,W)\), the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex \(i\in D\) is either on the tour or within a predetermined distance L to an arbitrary vertex \(j\in W\) on the tour, where \(D\subset V\),\(W\subset V\). In this paper, we propose the online CSP, where the salesman will encounter at most k blocked edges during the traversal. The edge blockages are real-time, meaning that the salesman knows about a blocked edge when it occurs. We present a lower bound \(\frac{1}{1 + (k + 2)L}k+1\) and a CoverTreeTraversal algorithm for online CSP which is proved to be \(k+\alpha \)-competitive, where \(\alpha =0.5+\frac{(4k+2)L}{OPT}+2\gamma \rho \), \(\gamma \) is the approximation ratio for Steiner tree problem and \(\rho \) is the maximal number of locations that a customer can be served. When \(\frac{L}{\texttt {OPT}}\rightarrow 0\), our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.     

An FPTAS for generalized absolute 1-center problem in vertex-weighted graphs

Given a vertex-weighted undirected connected graph \(G = (V, E, \ell , \rho )\), where each edge \(e \in E\) has a length \(\ell (e) > 0\) and each vertex \(v \in V\) has a weight \(\rho (v) > 0\), a subset \(T \subseteq V\) of vertices and a set S containing all the points on edges in a subset \(E' \subseteq E\) of edges, the generalized absolute 1-center problem (GA1CP), an extension of the classic vertex-weighted absolute 1-center problem (A1CP), asks to find a point from S such that the longest weighted shortest path distance in G from it to T is minimized. This paper presents a simple FPTAS for GA1CP by traversing the edges in \(E'\) using a positive real number as step size. The FPTAS takes \(O( |E| |V| + |V|^2 \log \log |V| + \frac{1}{\epsilon } |E'| |T| {\mathcal {R}})\) time, where \({\mathcal {R}}\) is an input parameter size of the problem instance, for any given \(\epsilon > 0\). For instances with a small input parameter size \({\mathcal {R}}\), applying the FPTAS with \(\epsilon = \Theta (1)\) to the classic vertex-weighted A1CP can produce a \((1 + \Theta (1))\)-approximation in at most O(|E| |V|) time when the distance matrix is known and \(O(|E| |V| + |V|^2 \log \log |V|)\) time when the distance matrix is unknown, which are smaller than Kariv and Hakimi’s \(O(|E| |V| \log |V|)\)-time algorithm and \(O(|E| |V| \log |V| + |V|^3)\)-time algorithm, respectively.     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).     

Semi-online scheduling with combined information on two identical machines in parallel

This paper is concerned with a semi-online scheduling problem with combined information on two identical parallel machines to minimize the makespan, where all the jobs have processing times in the interval \([1,\,t]\)  \((t\ge 1)\) and the jobs arrive in non-increasing order of their processing times. The objective is to minimize the makespan. For \(t\ge 1\), we obtain a lower bound \(\max _{N=1,2,3,\ldots }\left\{ \min \{\frac{4N+3}{4N+2}\,,\frac{Nt+N+1}{2N+1}\}\right\} \) and show that the competitive ratio of the \(LS\) algorithm achieves the lower bound.     

List 2-distance $$\varDelta +3$$-coloring of planar graphs without 4,5-cycles

Let \(\chi _2(G)\) and \(\chi _2^l(G)\) be the 2-distance chromatic number and list 2-distance chromatic number of a graph G, respectively. Wegner conjectured that for each planar graph G with maximum degree \(\varDelta \) at least 4, \(\chi _2(G)\le \varDelta +5\) if \(4\le \varDelta \le 7\), and \(\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1\) if \(\varDelta \ge 8\). Let G be a planar graph without 4,5-cycles. We show that if \(\varDelta \ge 26\), then \(\chi _2^l(G)\le \varDelta +3\). There exist planar graphs G with girth \(g(G)=6\) such that \(\chi _2^l(G)=\varDelta +2\) for arbitrarily large \(\varDelta \). In addition, we also discuss the list L(2, 1)-labeling number of G, and prove that \(\lambda _l(G)\le \varDelta +8\) for \(\varDelta \ge 27\).     

A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let \(G=(V,E)\) be a graph and \(\phi \) be a total \(k\)-coloring of \(G\) using the color set \(\{1,\ldots , k\}\). Let \(\sum _\phi (u)\) denote the sum of the color of the vertex \(u\) and the colors of all incident edges of \(u\). A \(k\)-neighbor sum distinguishing total coloring of \(G\) is a total \(k\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(\sum _\phi (u)\ne \sum _\phi (v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak first introduced this coloring and conjectured that \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph \(G\). In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with \(\Delta (G)\ge 7\). Moreover, we also show that \(\chi _{nsd}^{''}(G)\le \Delta (G)+2\) for planar graphs without intersecting triangles with \(\Delta (G) \ge 9\). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.     

$$(\alpha , \tau )$$-Monitoring for event detection in wireless sensor networks

Detecting abnormal events is one of the fundamental issues in wireless sensor networks (WSNs). In this paper, we investigate \((\alpha ,\tau )\)-monitoring in WSNs. For a given monitored threshold \(\alpha \), we prove that (i) the tight upper bound of \(\Pr [{S(t)} \ge \alpha ]\) is \(O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) \), if \(\mu (t) < \alpha \); and (ii) the tight upper bound of \(\Pr [{S(t)} \le \alpha ]\) is \(O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) \), if \(\mu (t) > \alpha \), where \(\Pr [X]\) is the probability of random event \(X,\, S(t)\) is the sum of the monitored area at time \(t,\, n\) is the number of the sensor nodes, \(sup\) is the upper bound of sensed data, \( \mu (t)\) is the expectation of \(S(t)\), and \(\ell ({x_1},{x_2}) = {x_1}\ln \left( {\frac{{{x_1}}}{{{x_2}}}} \right) + (1 - {x_1})\ln \left( {\frac{{1 - {x_1}}}{{1 - {x_2}}}} \right) \). An instant \((\alpha ,\tau )\)-monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous \((\alpha , \tau )\)-monitoring is investigated. We prove that the probability of false negative alarm is \(\delta \), if the sample size is Open image in new window for a given precision requirement, where Open image in new window is the Open image in new window fractile of a standard normal distribution. Finally, the performance of the proposed algorithms is validated through experiments.     

Convex median and anti-median at prescribed distance

The status of a vertex v in a connected graph G is the sum of the distances between v and all the other vertices of G. The subgraph induced by the vertices of minimum (maximum) status in G is called median (anti-median) of G. Let \(H=(G_1,G_2,r)\) denote a graph with \(G_1\) as the median and \(G_2\) as the anti-median of H, \(d(G_1,G_2)=r\) and both \(G_1\) and \(G_2\) are convex subgraphs of H. It is known that \((G_1,G_2,r)\) exists for every \(G_1\), \(G_2\) with \(r \ge \left\lfloor diam(G_1)/2\right\rfloor +\left\lfloor diam(G_2)/2\right\rfloor +2\). In this paper we show the existence of \((G_1,G_2,r)\) for every \(G_1\), \(G_2\) and \(r \ge 1\). We also obtain a sharp upper bound for the maximum status difference in a graph G.     

Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph \(G=(V_G, E_G)\) is defined as \(\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}\), where \(\delta _G(u)\) is the vertex degree of \(u\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as \(H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}\). In this paper, the extremal \(\bar{R}\)-values on several types of important graphs are considered. The graph with the maximum \(\bar{R}\)-value among all the simple connected graphs of diameter \(d\) is determined. Among the connected bipartite graphs of order \(n\), the graph with a given matching number (resp. vertex connectivity) having the maximum \(\bar{R}\)-value is characterized. Finally, sharp upper bounds on \(\bar{R}\)-value among all simple connected outerplanar (resp. planar) graphs are determined.     

Minimum 2-distance coloring of planar graphs and channel assignment

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).     

On (s,t)-relaxed L(2,1)-labeling of graphs

We initiate the study of relaxed \(L(2,1)\)-labelings of graphs. Suppose \(G\) is a graph. Let \(u\) be a vertex of \(G\). A vertex \(v\) is called an \(i\)-neighbor of \(u\) if \(d_G(u,v)=i\). A \(1\)-neighbor of \(u\) is simply called a neighbor of \(u\). Let \(s\) and \(t\) be two nonnegative integers. Suppose \(f\) is an assignment of nonnegative integers to the vertices of \(G\). If the following three conditions are satisfied, then \(f\) is called an \((s,t)\)-relaxed \(L(2,1)\)-labeling of \(G\): (1) for any two adjacent vertices \(u\) and \(v\) of \(G, f(u)\not =f(v)\); (2) for any vertex \(u\) of \(G\), there are at most \(s\) neighbors of \(u\) receiving labels from \(\{f(u)-1,f(u)+1\}\); (3) for any vertex \(u\) of \(G\), the number of \(2\)-neighbors of \(u\) assigned the label \(f(u)\) is at most \(t\). The minimum span of \((s,t)\)-relaxed \(L(2,1)\)-labelings of \(G\) is called the \((s,t)\)-relaxed \(L(2,1)\)-labeling number of \(G\), denoted by \(\lambda ^{s,t}_{2,1}(G)\). It is clear that \(\lambda ^{0,0}_{2,1}(G)\) is the so called \(L(2,1)\)-labeling number of \(G\). \(\lambda ^{1,0}_{2,1}(G)\) is simply written as \(\widetilde{\lambda }(G)\). This paper discusses basic properties of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of graphs. For any two nonnegative integers \(s\) and \(t\), the exact values of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of paths, cycles and complete graphs are determined. Tight upper and lower bounds for \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of complete multipartite graphs and trees are given. The upper bounds for \((s,1)\)-relaxed \(L(2,1)\)-labeling number of general graphs are also investigated. We introduce a new graph parameter called the breaking path covering number of a graph. A breaking path \(P\) is a vertex sequence \(v_1,v_2,\ldots ,v_k\) in which each \(v_i\) is adjacent to at least one vertex of \(v_{i-1}\) and \(v_{i+1}\) for \(i=2,3,\ldots ,k-1\). A breaking path covering of \(G\) is a set of disjoint such vertex sequences that cover all vertices of \(G\). The breaking path covering number of \(G\), denoted by \(bpc(G)\), is the minimum number of breaking paths in a breaking path covering of \(G\). In this paper, it is proved that \(\widetilde{\lambda }(G)= n+bpc(G^{c})-2\) if \(bpc(G^{c})\ge 2\) and \(\widetilde{\lambda }(G)\le n-1\) if and only if \(bpc(G^{c})=1\). The breaking path covering number of a graph is proved to be computable in polynomial time. Thus, if a graph \(G\) is of diameter two, then \(\widetilde{\lambda }(G)\) can be determined in polynomial time. Several conjectures and problems on relaxed \(L(2,1)\)-labelings are also proposed.     

Bounds on the domination number of a digraph

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S is adjacent from at least one vertex in S. The domination number of D, denoted by \(\gamma (D)\), is the minimum cardinality of a dominating set of D. The Slater number \(s\ell (D)\) is the smallest integer t such that t added to the sum of the first t terms of the non-increasing out-degree sequence of D is at least as large as the order of D. For any digraph D of order n with maximum out-degree \(\Delta ^+\), it is known that \(\gamma (D)\ge \lceil n/(\Delta ^++1)\rceil \). We show that \(\gamma (D)\ge s\ell (D)\ge \lceil n/(\Delta ^++1)\rceil \) and the difference between \(s\ell (D)\) and \(\lceil n/(\Delta ^++1)\rceil \) can be arbitrarily large. In particular, for an oriented tree T of order n with \(n_0\) vertices of out-degree 0, we show that \((n-n_0+1)/2\le s\ell (T)\le \gamma (T)\le 2s\ell (T)-1\) and moreover, each value between the lower bound \(s\ell (T)\) and the upper bound \(2s\ell (T)-1\) is attainable by \(\gamma (T)\) for some oriented trees. Further, we characterize the oriented trees T for which \(s\ell (T)=(n-n_0+1)/2\) hold and show that the difference between \(s\ell (T)\) and \((n-n_0+1)/2\) can be arbitrarily large. Some other elementary properties involving the Slater number are also presented.     

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).