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 \(F\cong K_{k+1,k+2}\).     

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.     

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.     

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).     

Nordhaus–Gaddum type result for the matching number of a graph

For a graph G, \(\alpha '(G)\) is the matching number of G. Let \(k\ge 2\) be an integer, \(K_{n}\) be the complete graph of order n. Assume that \(G_{1}, G_{2}, \ldots , G_{k}\) is a k-decomposition of \(K_{n}\). In this paper, we show that (1)

$$\begin{aligned} \left\lfloor \frac{n}{2}\right\rfloor \le \sum _{i=1}^{k} \alpha '(G_{i})\le k\left\lfloor \frac{n}{2}\right\rfloor . \end{aligned}$$

(2) If each \(G_{i}\) is non-empty for \(i = 1, \ldots , k\), then for \(n\ge 6k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n+k-1}{2}\right\rfloor . \end{aligned}$$

(3) If \(G_{i}\) has no isolated vertices for \(i = 1, \ldots , k\), then for \(n\ge 8k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n}{2}\right\rfloor +k. \end{aligned}$$

The bounds in (1), (2) and (3) are sharp. (4) When \(k= 2\), we characterize all the extremal graphs which attain the lower bounds in (1), (2) and (3), respectively.

     

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 \(\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)\).     

Even factors of graphs

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Favaron and Kouider (J Gr Theory 77:58–67, 2014) showed that if a simple graph G has an even factor, then it has an even factor F with \(|E(F)| \ge \frac{7}{16} (|E(G)| + 1)\). This ratio was improved to \(\frac{4}{7}\) recently by  Chen and Fan (J Comb Theory Ser B 119:237–244, 2016), which is the best possible. In this paper, we take the set of vertices of degree 2 (say \(V_{2}(G)\)) into consideration and further strengthen this lower bound. Our main result is to show that for any simple graph G having an even factor, G has an even factor F with \(|E(F)| \ge \frac{4}{7} (|E(G)| + 1)+\frac{1}{7}|V_{2}(G)|\).     

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 \(\chi _{2}(G)\le 5\Delta -7\).     

Near optimal algorithms for online weighted bipartite matching in adversary model

Bipartite matching is an important problem in graph theory. With the prosperity of electronic commerce, such as online auction and AdWords allocation, bipartite matching problem has been extensively studied under online circumstances. In this work, we study the online weighted bipartite matching problem in adversary model, that is, there is a weighted bipartite graph \(G=(L,R,E)\) and the left side L is known as input, while the vertices in R come one by one in an order arranged by the adversary. When each vertex in R comes, its adjacent edges and relative weights are revealed. The algorithm should irreversibly decide whether to match this vertex to an unmatched neighbor in L with the objective to maximize the total weight of the obtained matching. When the weights are unbounded, the best algorithm can only achieve a competitive ratio \(\varTheta \left( \frac{1}{n}\right) \), where n is the number of vertices coming online. Thus, we mainly deal with two variants: the bounded weight problem in which all weights are in the range \([\alpha , \beta ]\), and the normalized summation problem in which each vertex in one side has the same total weights. We design algorithms for both variants with competitive ratio \(\varTheta \left( \max \left\{ \frac{1}{\log \frac{\beta }{\alpha }},\frac{1}{n}\right\} \right) \) and \(\varTheta \left( \frac{1}{\log n}\right) \) respectively. Furthermore, we show these two competitive ratios are tight by providing the corresponding hardness results.     

On the performance of mildly greedy players in cut games

We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. (Theoret Comput Sci 438:13–27, 2012), where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than \(1+\epsilon \), for some given \(\epsilon \ge 0\). Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of \((1+\epsilon )\)-approximate pure Nash equilibria and \((1+\epsilon )\)-approximate best-responses, respectively. We first show that the \(\epsilon \)-approximate price of anarchy, that is the price of anarchy of \((1+\epsilon )\)-approximate pure Nash equilibria, is at least \(\frac{1}{2+\epsilon }\) and that this bound is tight for any \(\epsilon \ge 0\). Then, we evaluate the approximation ratio of the solutions achieved after a \((1+\epsilon )\)-approximate one-round walk starting from any initial strategy profile, where a \((1+\epsilon )\)-approximate one-round walk is a sequence of \((1+\epsilon )\)-approximate best-responses, one for each player. We improve the currently known lower bound on this ratio from \(\min \left\{ \frac{1}{4+2\epsilon },\frac{\epsilon }{4+2\epsilon }\right\} \) up to \(\min \left\{ \frac{1}{2+\epsilon },\frac{2\epsilon }{(1+\epsilon )(2+\epsilon )}\right\} \) and show that this is again tight for any \(\epsilon \ge 0\). An interesting and quite surprising consequence of our results is that the worst-case performance guarantee of the very simple solutions generated after a \((1+\epsilon )\)-approximate one-round walk is the same as that of \((1+\epsilon )\)-approximate pure Nash equilibria when \(\epsilon \ge 1\) and of that of subgame perfect equilibria (i.e., Nash equilibria for greedy players with farsighted, rather than myopic, rationality) when \(\epsilon =1\).     

An upper bound on the double Roman domination number

A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f : V \rightarrow \{0, 1, 2, 3\}\) having the property that if \(f(v) = 0\), then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with \(f(w)=3\), and if \(f(v)=1\), then vertex v must have at least one neighbor w with \(f(w)\ge 2\). The weight of a DRDF f is the value \(f(V) = \sum _{u \in V}f(u)\). The double Roman domination number \(\gamma _{dR}(G)\) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality \(\gamma _{dR}(G)\le \frac{6n}{5}\) and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, \(\gamma _{dR}(G)\le \frac{8n}{7}\).     

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 \(\varDelta \ge 27\).     

Hamiltonian numbers in oriented graphs

A hamiltonian walk of a digraph is a closed spanning directed walk with minimum length in the digraph. The length of a hamiltonian walk in a digraph D is called the hamiltonian number of D, denoted by h(D). In Chang and Tong (J Comb Optim 25:694–701, 2013), Chang and Tong proved that for a strongly connected digraph D of order n, \(n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor \), and characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor \). In the paper, we characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor -1\) and show that for any triple of integers n, k and t with \(n\ge 5\), \(n\ge k\ge 3\) and \(t\ge 0\), there is a class of nonisomorphic digraphs with order n and hamiltonian number \(n(n-k+1)-t\).     

Scheduling with interjob communication on parallel processors

Consider a scheduling problem in which a set of tasks needs to be scheduled on m parallel processors. Each task \(T_i\) consists of a set of jobs with interjob communication demands, represented by a weighted, undirected graph \(G_i\). The processors are assumed to be interconnected by a shared communication channel, which can be used by jobs to communicate among each other while being processed in parallel. In each time step, the scheduler assigns jobs to the processors and allows any processed job to use a certain capacity of the channel in order to satisfy (parts of) its communication demands to adjacent jobs processed in the same step. The goal is to find a schedule with minimum length in which the communication demands of all jobs are satisfied. We show that this problem is NP-hard in the strong sense even if the number of processors is constant and the underlying graph is a single path or a forest with arbitrary constant maximum degree. Consequently, we design and analyze approximation algorithms with asymptotic approximation ratio \(\min \{1.8, 1.5 \frac{m}{m-1}\}+1\) if the underlying graph G, the union of the \(G_i\), is a forest. For general graphs it is \(\min \left\{ 1.8, \frac{1.5m}{m-1}\right\} \cdot \left( \text {arb}(G) + \frac{5}{3}\right) \), where \(\text {arb}(G)\) denotes the arboricity of G.     

Bounding the total domination subdivision number of a graph in terms of its order

The total domination subdivision number \(\mathrm{sd}_{\gamma _{t}}(G)\) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that \(\mathrm{sd}_{\gamma_{t}}(G)\leq \lfloor\frac{2n}{3}\rfloor\) for any simple connected graph G of order n≥3 other than K ₄. We also determine all simple connected graphs G with \(\mathrm{sd}_{\gamma_{t}}(G)=\lfloor\frac{2n}{3}\rfloor\).     

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.     

Semi-online scheduling with known maximum job size on two uniform machines

In this paper, we investigate the semi-online scheduling problem with known maximum job size on two uniform machines with the speed ratio s≥1. The objective is to minimize the makespan. Two algorithms are presented, where the first is optimal for \(1\leq s\leq\sqrt{2}\), and the second is optimal for 1.559≤s≤2 and \(s\ge \frac{3+\sqrt{17}}{2}\). In addition, the improvement on lower bounds is made for \(2.     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.     

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.     

Upper bounds of proper connection number of graphs

A total weighting of a graph G is a mapping \(\phi \) that assigns a weight to each vertex and each edge of G. The vertex-sum of \(v \in V(G)\) with respect to \(\phi \) is \(S_{\phi }(v)=\sum _{e\in E(v)}\phi (e)+\phi (v)\). A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph \(G=(V,E)\) is called \((k,k')\)-choosable if the following is true: If each vertex x is assigned a set L(x) of k real numbers, and each edge e is assigned a set L(e) of \(k'\) real numbers, then there is a proper total weighting \(\phi \) with \(\phi (y)\in L(y)\) for any \(y \in V \cup E\). In this paper, we prove that for any graph \(G\ne K_1\), the Mycielski graph of G is (1,4)-choosable. Moreover, we give some sufficient conditions for the Mycielski graph of G to be (1,3)-choosable. In particular, our result implies that if G is a complete bipartite graph, a complete graph, a tree, a subcubic graph, a fan, a wheel, a Halin graph, or a grid, then the Mycielski graph of G is (1,3)-choosable.