A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph Kk n, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k − 2 vertices. Recently, the authors affirmed the conjecture. In the note we show that for every 2-coloring of Kk n, one can find two monochromatic paths of distinct col...

Let f(k) be the smallest integer such that every f(k)-chromatic digraph contains every oriented tree of order k. Burr proved f(k) ≤ (k − 1)2 in general, and conjectured f(k) = 2k − 2. Burr also proved that every (8k − 9)-chromatic digraph contains every antidirected tree. We improve both of Burr’s bounds. We show that f(k) ≤ k2/2 − k/2 + 1 and that every antidirected tree of order k is containe...

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, Sk(G) ≤ e(G) + k+1 2 for k = 1, . . . , n. We prove the conjecture for k = 2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with ...

There is a longstanding conjecture of Nussbaum, which asserts that every finite set in R on which a cyclic group of sup-norm isometries acts transitively contains at most 2 points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold more generally for abelian groups of sup-norm is...

Let [n]:={1,2,…,n} and let a k-set denote set of cardinality k. A family sets is union-closed (UC) if the union any two in also family. Frankl’s conjecture states that for nonempty UC F⊆2[n] such F≠{∅}, there exists an element i∈[n] contained at least half F, where 2[n] denotes power on [n]. The 3-sets Morris smallest number distinct (whose n-set) ensure satisfied (in contains them ⌊n/2⌋+1 all ...

We prove the Breuil–Mézard conjecture for 2-dimensional potentially Barsotti–Tate representations of the absolute Galois group GK , K a finite extension of Qp, for any p > 2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Qp is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serr...

We prove the Breuil–Mézard conjecture for 2-dimensional potentially Barsotti–Tate representations of the absolute Galois group GK , K a finite extension of Qp, for any p > 2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Qp is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serr...

Let di(m) denote the coefficients of the Boros-Moll polynomials. Moll’s minimum conjecture states that the sequence {i(i+1)(di (m)−di−1(m)di+1(m))}1≤i≤m attains its minimum at i = m with 2−2mm(m + 1) ( 2m m )2 . This conjecture is stronger than the log-concavity conjecture proved by Kauers and Paule. We give a proof of Moll’s conjecture by utilizing the spiral property of the sequence {di(m)}0≤...

In this note we formulate a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author’s diophantine conjecture for algebraic points of bounded degree. We also show that the latter conjecture implies the new conjecture. As with most of the author’s conjectures, this new conjecture stems from analogies with Nevanlinna theory. In this particular case the conjecture correspo...

Edmonds, Lov\'asz, and Pulleyblank showed that if a matching covered graph has nontrivial tight cut, then it also ELP-cut. Carvalho et al. gave stronger conjecture: cut $C$, ELP-cut does not cross $C$. Chen, al proof of the conjecture. This note is inspired by paper We give simplified conjecture, prove following result which slightly than $C$ $G$ an ELP-cut, there sequence $G_1=G, G_2,\ldots,G_...