Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of verte...Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.展开更多
Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two positive integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). Then a (g, f)-factor of G ...Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two positive integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤ dH(x) 5 f(x) for each x ∈ V(H). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F1, F2,…… ， Fm } and H be a factorization and a subgraph of G, respectively. If F, 1 ≤ i ≤ m， has exactly one edge in common with H, then it is said that ■ is orthogonal to H. It is proved that every bipartite (mg + m - 1, mf - m + 1 )-graph G has a (g, f)-factorization orthogonal to k vertex disjoint m-subgraphs of G if 2-k ≤ g(x) for all x ∈ V(G). Furthermore, it is showed that the results in this paper are best possible.展开更多
The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we character...The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we characterize the graphs with the maximum and second-maximum Wiener indices among all the graphs in G＊c, respectively.展开更多
The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that ther...The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.展开更多
We propose a pilot domain non-orthogonal multiple access(NOMA)for uplink massive devices grant-free random access scenarios in massive multiple-input multiple-output(MIMO)maritime communication systems.These scenarios...We propose a pilot domain non-orthogonal multiple access(NOMA)for uplink massive devices grant-free random access scenarios in massive multiple-input multiple-output(MIMO)maritime communication systems.These scenarios are characterized by numerous devices with sporadic access behavior,and therefore only a subset of them are active.Due to massive potential devices in the network,it is infeasible to assign a unique orthogonal pilot to each device in advance.In such scenarios,pilot decontamination is a crucial problem.In this paper,the devices are randomly assigned non-orthogonal pilots which are constructed by a linear combination of some orthogonal pilots.We show that a bipartite graph can conveniently describe the interference cancellation(IC)processes of pilot decontamination.High spectrum efficiency(SE)and low outage probability can be obtained by selecting the numbers of orthogonal pilots according to the given probability distribution.Numerical evaluatioDs show that the proposed pilot domain NOMA decreases the outage probability from 20%to 2 e-12 at the SE of 4 bits/s/Hz for a single device,compared to the conventional method of slotted ALOHA with 1024 antennas at the BS,or increases the spectrum efficiency from 1.2 bits/s/Hz to 4 bit/s/Hz at the outage probability of2 e-12 in contrast with the Welch bound equality(WBE)non-orthogonal pilots.展开更多
Top-k ranking of websites according to traffic volume is important for Internet Service Providers(ISPs) to understand network status and optimize network resources. However, the ranking result always has a big deviati...Top-k ranking of websites according to traffic volume is important for Internet Service Providers(ISPs) to understand network status and optimize network resources. However, the ranking result always has a big deviation with actual rank for the existence of unknown web traffic, which cannot be identified accurately under current techniques. In this paper, we introduce a novel method to approximate the actual rank. This method associates unknown web traffic with websites according to statistical probabilities. Then, we construct a probabilistic top-k query model to rank websites. We conduct several experiments by using real HTTP traffic traces collected from a commercial ISP covering an entire city in northern China. Experimental results show that the proposed techniques can reduce the deviation existing between the ground truth and the ranking results vastly. In addition, we find that the websites providing video service have higher ratio of unknown IP as well as higher ratio of unknown traffic than the websites providing text web page service. Specifically, we find that the top-3 video websites have more than 90% of unknown web traffic. All these findings are helpful for ISPs understanding network status and deploying Content Distributed Network(CDN).展开更多
The total chromatic number xT（G） of a graph G is the minimum number of colors needed to color the elements（vertices and edges） of G such that no adjacent or incident pair of elements receive the same color, G is c...The total chromatic number xT（G） of a graph G is the minimum number of colors needed to color the elements（vertices and edges） of G such that no adjacent or incident pair of elements receive the same color, G is called Type 1 if xT（G） =△（G）＋1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.展开更多
基金Supported by the National Natural Science Foundation of China（61163037, 61163054, 11261046, 61363060）
文摘Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.
基金This work was supported by NNSF. RFDP and NNSF of shandong province(Z2000A02 ).
文摘Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two positive integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤ dH(x) 5 f(x) for each x ∈ V(H). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F1, F2,…… ， Fm } and H be a factorization and a subgraph of G, respectively. If F, 1 ≤ i ≤ m， has exactly one edge in common with H, then it is said that ■ is orthogonal to H. It is proved that every bipartite (mg + m - 1, mf - m + 1 )-graph G has a (g, f)-factorization orthogonal to k vertex disjoint m-subgraphs of G if 2-k ≤ g(x) for all x ∈ V(G). Furthermore, it is showed that the results in this paper are best possible.
文摘The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we characterize the graphs with the maximum and second-maximum Wiener indices among all the graphs in G＊c, respectively.
文摘The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.
基金supported by Key R&D Program of China under Grant 2018YFB1801102National Natural Science Foundation of China(U1736108)+1 种基金Foundation for Innovative Research Groups of the National Natural Science Foundation of China(61621091)Tsinghua University Initiative Scientific Research Program 20193080005。
文摘We propose a pilot domain non-orthogonal multiple access(NOMA)for uplink massive devices grant-free random access scenarios in massive multiple-input multiple-output(MIMO)maritime communication systems.These scenarios are characterized by numerous devices with sporadic access behavior,and therefore only a subset of them are active.Due to massive potential devices in the network,it is infeasible to assign a unique orthogonal pilot to each device in advance.In such scenarios,pilot decontamination is a crucial problem.In this paper,the devices are randomly assigned non-orthogonal pilots which are constructed by a linear combination of some orthogonal pilots.We show that a bipartite graph can conveniently describe the interference cancellation(IC)processes of pilot decontamination.High spectrum efficiency(SE)and low outage probability can be obtained by selecting the numbers of orthogonal pilots according to the given probability distribution.Numerical evaluatioDs show that the proposed pilot domain NOMA decreases the outage probability from 20%to 2 e-12 at the SE of 4 bits/s/Hz for a single device,compared to the conventional method of slotted ALOHA with 1024 antennas at the BS,or increases the spectrum efficiency from 1.2 bits/s/Hz to 4 bit/s/Hz at the outage probability of2 e-12 in contrast with the Welch bound equality(WBE)non-orthogonal pilots.
基金supported by 111 Project of China under Grant No.B08004
文摘Top-k ranking of websites according to traffic volume is important for Internet Service Providers(ISPs) to understand network status and optimize network resources. However, the ranking result always has a big deviation with actual rank for the existence of unknown web traffic, which cannot be identified accurately under current techniques. In this paper, we introduce a novel method to approximate the actual rank. This method associates unknown web traffic with websites according to statistical probabilities. Then, we construct a probabilistic top-k query model to rank websites. We conduct several experiments by using real HTTP traffic traces collected from a commercial ISP covering an entire city in northern China. Experimental results show that the proposed techniques can reduce the deviation existing between the ground truth and the ranking results vastly. In addition, we find that the websites providing video service have higher ratio of unknown IP as well as higher ratio of unknown traffic than the websites providing text web page service. Specifically, we find that the top-3 video websites have more than 90% of unknown web traffic. All these findings are helpful for ISPs understanding network status and deploying Content Distributed Network(CDN).
文摘The total chromatic number xT（G） of a graph G is the minimum number of colors needed to color the elements（vertices and edges） of G such that no adjacent or incident pair of elements receive the same color, G is called Type 1 if xT（G） =△（G）＋1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.
