Due to the ubiquitous coverage and seamless connectivity, cellular systems are very promising to support Machineto-Machine (M2M) communications. But, all of the cellular networks are designed and optimized for Human-to-Human (H2H) or Human-to-Machine (H2M) communications and therefore facing several challenges due to incorporation of M2M communications. One of such challenges is efficient resource allocation to M2M applications without affecting or least affecting H2H applications. In order to address this challenge, we need application specific priority based scheduling algorithms in which based on the QoS of the application, radio resources are allocated.In this paper, we have classified and prioritized all H2H and M2M flows based on their QoS requirements. Resources are allocated first to higher priority classes and in a given class, they are allocated to H2H flows first. In order to ensure the QoS of H2H flows, a threshold is kept on the maximum number of radio resource blocks to be assigned to M2M flows in a scheduling interval. Performance of the proposed scheduling algorithm is evaluated using various metrics such as system throughput and average utility per class and compared against existing scheduling schemes.

Let G be a finite simple graph on n vertices. Let JG ⊂ K[x1, . . . , xn] be the cover ideal of G. In this article, we obtain the graded minimal free resolution and the Castelnuovo-Mumford regularity of J s G for all s ≥ 1 for certain multipartite graphs G.G 1 for some classes of multipartite graphs, thereby obtaining a precise expression for reg(J s G ). The paper is organized as follows. In Section 2, we collect the preliminaries required for the rest of the paper. We study the resolution of powers of cover ideals of certain bipartite graphs in Section 3. If G is a complete bipartite graph, then J G is a regular sequence and hence the minimal graded free resolution of J s G can be obtained from [9, Theorem 2.1]. It can be seen that, in this case the index of stability, s 0 = 1 and the constant term is one less than the size of the minimum vertex cover. We then move on to study some classes of bipartite graphs which are not complete. We obtain the resolution and precise expressions for the regularity of powers of cover ideals of certain bipartite graphs.Section 4 is devoted to the study of resolution and regularity of powers of cover ideals of certain complete multipartite graphs. When G is the cycle of length three or the complete graph on 4 vertices, we describe the graded minimal free resolution of J s G for all s ≥ 1. This allow us to compute the Betti numbers, Hilbert series and the regularity of J s G for all s ≥ 1. As a consequence, for cover ideals of complete tripartite and 4-partite graphs, we obtain precise expressions for the Betti numbers and the regularity of J s G . We conclude our article with a conjecture on the resolution of J s G for all s ≥ 1, where G is a complete multipartite graph. Acknowledgements: Part of the work was done while the second author was visiting Indian Institute of Technology Madras. He would like to thank IIT Madras for their hospitality during the visit. All our computations were done using Macaulay 2, [8].

ABSTRACT. Let S = K[x 1 , . . . , x n ] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f 1 , f 2 , . . . , f k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and Valla [9] concerning Koszul property of the diagonal subalgebras associated to I. Each such subalgebra has the form K[(I e ) ed+c ], where c, e ∈ N. For k = 3, we extend [9, Corollary 6.10] by proving that K[(I e ) ed+c ] is Koszul as soon as c ≥ d2 and e > 0. We also extend [9, Corollary 6.10] in another direction by replacing the polynomial ring with a Koszul ring.

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