The game of Cops and Robbers is a turn-based game played on a graph, in which a team of cops attempts to capture a robber. The cop number of a graph is the minimum number of cops required to capture the robber. In this paper, we study two variants of this game: the surrounding cops and robber game, and a version involving cops and an insightful robber. Burgess et al. (Discrete Appl. Math. 285 (2020), 552–566) established a lower bound for the surrounding cop number of the strong product Km,n⊠Km,n. We close the gap by determining the exact value for all positive integers m and n. In addition, we obtain tight results for the minimum number of cops required to capture an insightful robber on the strong products Pm⊠Cn and Cm⊠Cn. Our results extend a result of Huggan and Nowakowski (Discrete Appl. Math. 295 (2021), 112–119), previously obtained for Pm⊠Pn.
Let G=(V(G),E(G)) be a graph, and let S⊆V(G) with |S|≥2. Let a,b:S→N be functions. We show that if dG(x)≥a(x)+b(x)+1 for each x∈S, then there exists a partition (A,B) of V(G) such that A∩S≠0̸, B∩S≠0̸, and for each x∈A∩S, dG[A](x)≥a(x), while for each x∈B∩S, dG[B](x)≥b(x). This theorem generalizes the classical decomposing theorem of Stiebitz.
In this article, we present the first systematic approach for the decomposition of s-plateaued functions f∈Bn. It is well-known that bent functions can be decomposed into two (disjoint spectra) semi-bent functions, whereas their 4-decomposition (so that f=f1||f2||f3||f4) gives precisely three cases. However, for s-plateaued functions on F2n with spectra Wf(a)∈{0,±2(n+s)/2} the situation is quite different. We provide a full characterization of both 2-decomposition and 4-decomposition of any s-plateaued function. In the latter case, the Walsh spectral values of fi belong to {0,±2p−2,±2p−1,±3⋅2p−2,±2p}, where p=n+s2. This implies that the restrictions fi can have at most 9 different spectral values, which is also demonstrated by specifying a semi-bent function f(x,y)=x⋅ϕ(y)+h(y) in the GMM class having this type of 4-decomposition. In contrast to bent functions, mixed 4-decompositions (thus fi are not of the same type) are feasible, demonstrating a greater variety of different scenarios. We also provide sufficient and necessary conditions for the design methods of s-plateaued functions when the subfunctions fi belong to GMM. In particular, when ϕi are 2t-to-1 non-injective mappings, we demonstrate that f=f1||f2||f3||f4 may not admit M-subspaces of large dimension. This feature, assuming that fi are disjoint semi-bent functions, can be further utilized in the design of bent functions outside the Maiorana-McFarland (M) class. This possibility is demonstrated in the last part of this article, where two instances of bent functions outside the completed class M# are presented.
We say that a family of graphs G admits a rainbow k-factor if there exists a k-regular spanning subgraph H of the common vertex set V such that each edge of H is drawn from a distinct graph in G. In this paper, we derive several spectral sufficient conditions under which a family of graphs admits a rainbow substructure. Specifically, let G={G1,G2,…,Gs+1} be a collection of graphs sharing the same vertex set V={1,2,…,n}, and let ρα(Gi) denote the spectral radius of the Aα-matrix of Gi, where Aα(Gi)=αD(Gi)+(1−α)A(Gi) for α∈[0,1). First, we prove that if, for all i∈{1,2,…,s+1} and for any α∈[0,1), ρα(Gi)≥max2s,nα+s−1+(nα+s−1)2−4sα(s−1)+4s(n−s)(1−2α)2, then G contains a rainbow matching, i.e., a set of s+1 pairwise disjoint edges {e1,e2,…,es+1} with ei∈E(Gi), unless all Gi are identical and each equals either K2s+1∪(n−2s−1)K1 or Ks∨(n−s)K1. Second, we establish a spectral sufficient condition — expressed in terms of the Aα-spectral radii of the constituent graphs — that guarantees the existence of a rainbow k-factor in G.
