The chromatic number of the categorical product of two n-tournaments can be strictly smaller than n. We show that min{χ(S × T ) : S and T are n-tournaments} is asymptotically equal to λn, where 12 ≤ λ ≤ 2 3 .

We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.

Analogues and variations of Ulam’s notorious graph reconstruction conjecture have been studied for a variety of combinatorial objects, for instance words (see Schützenberger and Simon [2, Theorem 6.2.16]), permutations (see Raykova [4] and Smith [5]), and compositions (see Vatter [6]), to name a few. In answer to Cameron’s query [1] about the partition context, Pretzel and Siemons [3] proved th...

Starting new businesses is important for the social and economic development of each country (Esfandiar et al., 2019; Fayolle & Liñán, 2014). However, research on starting a business has so far been more about clarifying influence factors intention (Duong, 2022; Ghosh, Loan 2021; Vasilev, Vuong 2020), there have not many studies entrepreneurial success—the determinant meaning (Wang 2023; We...

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

Around 1960, Markus & Yamabe [1] conjectured that, for each n ≥ 2, every rest point x0 of a nonlinear, class C1, n-dimensional system of differential equations ẋ = F (x) is globally asymptotically stable if the eigenvalues of the jacobian matrix F (x) have negative real parts at every point x in R. At that time, Olech [2] connected this problem with the question of injectivity of the mapping x ...

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on n vertices is at most ⌊ n2 4 ⌋ and the extremal graph is the complete bipartite graph Kb 2 c,d 2 e. In the series papers [8–10], the Murty-Simon Conjecture stated by Haynes et al. is not th...

A distinguishing r-labeling of a digraph G is a mapping λ from the set of vertices of G to the set of labels {1, . . . , r} such that no nontrivial automorphism of G preserves all the labels. The distinguishing number D(G) of G is then the smallest r for which G admits a distinguishing r-labeling. From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups, Can. J....

Let G = (V,E) be a simple graph. A subset Dof V (G) is a (k, r)dominating set if for every vertexv ∈ V −D, there exists at least k vertices in D which are at a distance utmost r from v in [1]. The minimum cardinality of a (k, r)-dominating set of G is called the (k, r)-domination number of G and is denoted by γ(k,r)(G). In this paper, minimal (k, r)dominating sets are characterized. It is prove...

in this paper, we show that for any finite order entire function $f(z)$, the function of the form $f(z)^{n}[f(z+c)-f(z)]^{s}$ has no nonzero finite picard exceptional value for all nonnegative integers $n, s$ satisfying $ngeq 3$, which can be viewed as a different result on hayman conjecture. we also obtain some uniqueness theorems for difference polynomials of entire functions sharing one comm...