A k-edge-weighting of a graph G is a mapping w : E(G) → {1, 2, . . ., k}. An edge-weighting w induces a vertex coloringfor any edge uv. The current paper studies the parameter µ(G), which is the minimum k for which G has a vertex-coloring k-edgeweighting. Exact values of µ(G) are determined for several classes of graphs, including trees and r-regular bipartite graph with r ≥ 3.

In this paper we consider a fundamental problem in the area of viral marketing, called TARGET SET SELECTION problem. We study the problem when the underlying graph is a block-cactus graph, a chordal graph or a Hamming graph. We show that if G is a block-cactus graph, then the TARGET SET SELECTION problem can be solved in linear time, which generalizes Chen's result [2] for trees, and the time complexity is much better than the algorithm in [1] (for bounded treewidth graphs) when restricted to block-cactus graphs. We show that if the underlying graph G is a chordal graph with thresholds θ(v) ≤ 2 for each vertex v in G, then the problem can be solved in linear time. For a Hamming graph G having thresholds θ(v) = 2 for each vertex v of G, we precisely determine an optimal target set S for (G, θ). These results partially answer an open problem raised by Dreyer and Roberts [3].

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With growing popularity of cloud storage, the number of users of outsourcing data to cloud servers has increased dramatically. On the one hand, the rapidly increasing volume of data in the cloud is accompanied by a lot of data duplication. On the other hand, the cloud server stores only a unique copy of outsourced data in deduplication cloud storage system and the corruption or missing of the unique copy may bring immeasurable loss. Therefore, the file deduplication and integrity auditing are very important and how to securely and efficiently achieve them simultaneously needs to be settled urgently in academia and industry. In this paper, we propose a confidentiality-preserving deduplication cloud storage with public cloud auditing (CPDA). Firstly, our CPDA scheme achieves secure file deduplication on encrypted file, which supports public integrity auditing for the unique copy in the deduplication cloud storage system. Particularly, our CPDA scheme also realizes secure authentication tag deduplication. Secondly, our CPDA scheme utilizes the convergent encryption and random masking techniques to ensure data confidentiality during the file deduplication and integrity auditing process. Thirdly, our scheme not only supports each data owner to independently launch the integrity auditing of their own files, but also supports cloud server to periodically delegate the third party auditor to concurrently handle multiple auditing tasks to ensure the integrity of the outsourced files. Finally, the security of our scheme is formally proved and its performance is confirmed by numerical analyses and simulation experiments. INDEX TERMS Cloud storage, secure deduplication, public cloud auditing, batch verification.