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Inside a two-dimensional region ("cake"), there are m nonoverlapping tiles of a certain kind ("toppings"). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same "certain kind," such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks. [ABSTRACT FROM AUTHOR]

We prove an upper bound on the Shannon capacity of a graph via a linear programming variation. We show that our bound can outperform both the Lov\'asz theta number and the Haemers minimum rank bound. As a by-product, we also obtain a new upper bound on the broadcast rate of index coding. [ABSTRACT FROM AUTHOR]

We first consider the following problem. We are given a fixed perfect matching M of [n] and we add random edges one at a time until there is a Hamilton cycle containing M. We show that with high probability (w.h.p.) the hitting time for this event is the same as that for the first time there are no isolated vertices in the graph induced by the random edges. We then use this result for the following problem. We generate random edges and randomly color them black or white. A path/cycle is said to be zebraic if the colors alternate along the path. We show that w.h.p. the hitting time for a zebraic Hamilton cycle coincides with every vertex meeting at least one edge of each color. We then consider some related problems and (partially) extend our results to multiple colors. We also briefly consider directed versions. [ABSTRACT FROM AUTHOR]

A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e., finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP ⊆ coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of constraints that separate kernelizable from nonkernelizable variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a global relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a local relaxation parameter, and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t ≥ 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2c² k7+c log³ k) whenever c ∈ o(√k). Thus the kernel size varies from being subexponential when c ∈ o(√k), to quasi-polynomial when c ∈ o(logℓ k), ℓ ∈ R+, and polynomial when c ∈ O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t ∈ O(k∈) for any 0≤∈< 1, unless NP 埦 coNP/poly. [ABSTRACT FROM AUTHOR]

Gyárfás, Győri, and Simonovits [J. Comb., 7 (2016), pp. 205{216] proved that if a 3-uniform hypergraph with n vertices has no linear cycles, then its independence number α ≥ 2n/5. The hypergraph consisting of vertex disjoint copies of a complete hypergraph K3/5 on five vertices shows that equality can hold. They asked whether this bound can be improved if we exclude K3/5 as a subhypergraph and whether such a hypergraph is 2-colorable. In this paper, we answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph doesn't contain K3 5 as a subhypergraph, then it is 2-colorable. This result clearly implies that its independence number α ≥ [n/2]. We show that this bound is sharp. Gyárfás, Gy}ori, and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n - 2 when n ≥ 10. [ABSTRACT FROM AUTHOR]