The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph [Formula: see text] is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. In this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.

In 1973, Erdős et al. introduced the anti-Ramsey number for a graph G in Kn, which is defined to be the maximum number of colors in an edge-coloring of Kn which does not contain any rainbow G. This is always regarded as one of rainbow generalizations of the classic Ramsey theory. Since then the anti-Ramsey numbers for several special graph classes in complete graphs have been determined. Also, the researchers generalized the host graph for the anti-Ramsey number from the complete graph to general graphs, including bipartite graphs, complete split graphs, planar graphs, and so on. In this paper, we study the anti-Ramsey number of matchings in the complete split graph. Since the complete split graph contains the complete graph as a subclass, the results in this paper cover the previous results about the anti-Ramsey number of matchings in the complete graph.

Graph coloring problem and problem on the existence of paths and cycles have always been popular topics in graph theory. The problem on the existence of rainbow paths and rainbow cycles in edge colored graphs, as an integration of them, was well studied for a long period. In this survey, we will review known results on this subject. Because of the relationship between cycles and paths, we will review results on the existence of rainbow cycles (including rainbow Hamilton cycles, long rainbow cycles and rainbow cycles with given length) first, and then long rainbow paths (including rainbow Hamilton paths and other long rainbow paths).

Coupon coloring is a new coloring which has many applications. A k-coupon coloring of a graph G is a k-coloring of G by colors [k]={1,2,…,k} such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum integer k for which a k-coupon coloring exists is called the coupon coloring number of G, and it is denoted by χc(G). In this paper, we studied the coupon coloring of cographs, which are graphs that can be generated from the single vertex graph K1 by complementation and disjoint union, and have applications in many interesting problems. We use the cotree representation of a cograph to give a polynomial time algorithm to color the vertices of a cograph, and then prove that this coloring is a coupon coloring with maximum colors, hence get the coupon coloring numbers of the cograph.

This paper proposes a new nonlinear tracking control scheme with simultaneous unknown mass identification for magnetic suspension systems. Specifically, an amplitude-saturated adaptive control law is developed to achieve stable tracking and accurately estimate the unknown suspended mass simultaneously. The stability is assured with rigorous Lyapunov-based analysis. As far as we know, this is the first continuous control method for magnetic suspension systems with unknown levitated ball mass and actuator saturation, yielding an asymptotic result to achieve simultaneous tracking control and mass identification. Through hardware experiments, we verify the performance of the proposed method and compare it with existing methods.

Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v to be the number of different colors that are used on edges incident to v, and denote it by \(d^c(v)\). In a previous paper, we showed that if \(d^c(v)\ge k\) (color degree condition) for every vertex v of G, then G has a rainbow path of length at least \(\lceil (k+1)/2\rceil \). Later, in another paper we first showed that if \(k\le 7\), G has a rainbow path of length at least \(k-1\), and then, based on this we used induction on k and showed that if \(k\ge 8\), then G has a rainbow path of length at least \(\lceil (3k)/5\rceil +1\). In 2010, Gyárfás and Mhalla showed that in any proper edge-colored complete graph \(K_n\), there is a rainbow path with no less than \((2n+1)/3\) vertices. In the present paper, by using a simpler approach we further improve the result by showing that if \(k\ge 8\), G has a rainbow path of length at least \(\lceil (2k)/3\rceil +1\).

Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v be the number of different colors that are used on the edges incident to v, and denote it to be d^c(v). It was shown that if d^c(v)≥ k for every vertex v of G, then G has a rainbow path of length at least min{⌈2k+1/3⌉,k-1}. In the present paper, we consider the properly edge-colored complete graph K_n only and improve the lower bound of the length of the longest rainbow path by showing that if n≥ 20, there must have a rainbow path of length no less than 3/4n-1/4√(n/2-39/11)-11/16.

This paper studies the problem of identifying rumor source in online social networks in which the spread of information follows the popular Independent Cascade model. In the absence of text information, we develop a monitor based approach to evaluate how likely that a piece of information is actually a rumor. Given the underlying social network structure, a number of monitor nodes are injected into the network whose job is to report the data they receive. Based on observing which of monitors received the information and which did not, we propose a polynomial time algorithm to compute rumor quantifier, a reachability based score for ranking the importance of nodes as the rumor source. Extensive simulation results have shown that, with a reasonable number of monitor nodes and appropriate monitor deployment, our rumor source detection algorithm can recognize rumor source effectively and efficiently.

A boy wants to make friends with a pretty girl. He feels that he may get rejected if he invites her directly. In this situation, what he could do is to influence the girl's friends. Similar situations may occur in social activities. Based on this background, we formulate a new optimization problem, the Target Influence Maximization (TIM) problem and show that this problem can be solved in polynomial-time in networks with no directed cycles. Motivated by this, we study a special strategy to construct solutions for TIM, i.e., The Target Influence Maximization through Sub graph without Directed Cycle (TIMSDC). Two polynomial-time approximation algorithms are designed for TIMSDC. Through extensive experiments on real-world data sets, we demonstrate that our algorithms work efficiently and outperform existing methods.

The interaction between individuals are usually modeled as weighted edges in a social network. This information, however, is often unavailable in practice. On the other hand, information diffusion process upon the underlying network is observable. Hence sophisticated algorithm is needed to infer the edge set and edge weights from observed cascade set. To deal with this problem, we derive the likelihood of a given network generating a cascade set. With this likelihood, we design a distributed algorithm named Net Win that first calculates the optimal edge weights by maximizing likelihood and then sparsifies the result of optimization by a novel post-processing algorithm. In experimental results, Net Win infers various networks with high accuracy and outperforms other state-of-the-art algorithms in almost all cases.

ABSTRACT Combining cooperative diversity, truncated automatic repeat request scheme and distributed energy‐aware routing strategy, a novel cooperative routing algorithm adopting decode‐and‐forward fashion is proposed to maximise the lifetime of wireless sensor networks from the cross‐layer design perspective. In this algorithm, the transmission power is optimally allocated while satisfying the per‐link symbol error rate or the end‐to‐end throughput requirement. The average total consumed power weighted by the normalised remaining energy of every participating node is used as the link cost to avoid the overuse of certain nodes with extremely little energy. With the use of the traditional distributed shortest path algorithm, the best route that includes a cascade of single‐relay building blocks is constructed with polynomial complexity. In contrast to the non‐cooperative routing schemes, this cooperative routing algorithm can significantly prolong the network lifetime and improve the energy efficiency by reducing the total network residual energy. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with k-good colorings, i.e., minimum color degree at least k. For the heterochromatic triangle free graphs K-n, we obtain that for every vertex v is an element of V (K-n), K-n has a heterochromatic v-path of length at least d(c)(v); whereas for the heterochromatic triangle free graphs G we show that if, for any vertex v is an element of V(G), d(c)(v)>= k >= 6, then G has a heterochromatic path of length at least 3k/4.

Cooperative diversity has recently been proposed as a promising technology to achieve spatial diversity in wireless networks. In this paper, we analyze the performance of SNR-based hybrid decode-amplify-forward (HDAF) relaying cooperative diversity networks over independent non-identical flat Rayleigh fading channels with maximum ratio combining (MRC) technique. Closed-form expressions for the outage and bit error probability of the HDAF relaying scheme are derived. Computer simulations are carried out to illustrate and validate the correctness of analytical results. Besides, the impacts of the SNR threshold and relay location on the performances of outage and bit error probability are investigated.

Three new power allocation schemes for a typical amplify-and-forward (AF) cooperative system over independent non-identical Nakagami-m fading channels are proposed in this paper. By analyzing the upper bound of the instantaneous SNR (signal-to-noise-ratio) at the destination using the inequity expression between arithmetic mean and geometric mean, we derive the average SNR and SER (symbol-error-rate) for M-ary phase-shift-keying (M-PSK) modulation scheme. Moreover, we seek to find optimal power allocation strategies that maximize the upper bound of average SNR or minimize the upper bound of average SER at the receiver side with total and single-link power constraints. Through carefully observing the expression structure of instantaneous SNR, an equivalent circuit form is abstracted. Meanwhile, a new power allocation scheme based on the equivalent circuit form is investigated. Monte Carlo simulation results are included to verify the effectiveness and superiority of these proposed new power allocation strategies.

Cooperative diversity has recently been proposed as a promising technology to achieve spatial diversity in wireless networks. In, the performance of incremental amplify-and-forward (AF) relaying cooperative diversity networks is analyzed. In this paper, a new relaying scheme in conjunction with incremental decode-and-forward (DF) relaying and selective DF relaying strategies, termed incremental-selective DF relaying, is proposed and analyzed. Closed-form expressions for error probability of both the incremental-selective DF relaying scheme and the incremental DF relaying scheme are derived. The effect of signal-to-noise ratio (SNR) threshold on the error probability is discussed. Moreover, results show that incremental-selective DF relaying scheme outperforms incremental DF relaying scheme for all the cases we investigate.

An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we give an explicit formula for the heterochromatic tree partition number of an r-edge-colored complete bipartite graph Km,n.

Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lfloor{2s+4\over5}\rfloor$. In the present paper, we prove that $G$ has a heterochromatic path of length at least $\lceil{s+1\over2}\rceil$, and give examples to show that the lower bound is best possible in some sense.

This paper has been withdrawn by the author(s), due an error in the proof.

Let $G$ be an edge-colored graph. A heterochromatic path of $G$ is such a path in which no two edges have the same color. $d^c(v)$ denotes the color degree of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil{k+1\over 2}\rceil$. It is easy to see that if $k=1,2$, $G$ has a heterochromatic path of length at least $k$. Saito conjectured that under the color degree condition $G$ has a heterochromatic path of length at least $\lceil{2k+1\over 3}\rceil$. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito's conjecture, we can show in this paper that if $3\leq k\leq 7$, $G$ has a heterochromatic path of length at least $k-1,$ and if $k\geq 8$, $G$ has a heterochromatic path of length at least $\lceil{3k\over 5}\rceil+1$. Actually, we can show that for $1\leq k\leq 5$ any graph $G$ under the color degree condition has a heterochromatic path of length at least $k$, with only one exceptional graph $K_4$ for $k=3$, one exceptional graph for $k=4$ and three exceptional graphs for $k=5$, for which $G$ has a heterochromatic path of length at least $k-1$. Our experience suggests us to conjecture that under the color degree condition $G$ has a heterochromatic path of length at least $k-1$.