Due to the limitation of energy resources, energy efficiency is a key issue in wireless sensor networks (WSNs). Clustering is proved to be an important way to realize hierarchical topology control, which can improve the scalability and prolong the lifetime of wireless sensor networks. In this paper, an energy-driven unequal clustering protocol (EDUC) for heterogeneous wireless sensor networks is proposed. EDUC includes an unequal clustering algorithm and an energy-driven adaptive cluster head rotation method. The unequal size of clusters can balance the energy consumption among clusters, and the energy-driven cluster head rotation method can achieve the balance of energy consumption among nodes within a cluster, which reduces the waste of energy. Simulation experiments show that EDUC balances the energy consumption well among the cluster heads and prolongs the network lifetime.
For any even integer k and any integer i,we prove that a (kr+i)-regular multigraphcontains a k-factor if it contains no more than kr-(3k)/2+i+2 cut edges,and this result is the bestpossible to guarantee the existence of k-factor in terms of the number of cut edges.We further give acharacterization for k-factor free regular graphs.
A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a (G) Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a (G) max{2Δ(G) + 2, Δ(G) + 22} if g(G) 3, a (G) Δ(G) + 2 if g(G) 5, a (G) Δ(G) + 1 if g(G) 7, and a (G) = Δ(G) if g(G) 16 and Δ(G) 3. For series-parallel graphs G, we have a (G) Δ(G) + 1.
