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...

‎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...

In this paper, the authors consider argument mining as the task of building a formal representation for an argumentative piece of text. Their goal is to provide a critical survey of the literature on both the resulting representations (i.e., argument diagramming techniques) and on the various aspects of the automatic analysis process. For representation, the authors also provide a synthesized p...

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...