Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when $k\le 4$ or $k\ge \frac{n?2}{3}$. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when $k=5$ or $k\ge \frac{n+2}{5}$. This completely resolves two conjectures by Gutiérrez in the affirmative.     

On the full Brouwer's Laplacian spectrum conjecture

Let G be a simple connected graph and let $S_k(G)$ be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that $S_k(G)\le e(G)+\left(\begin{array}{c}k+1\\ 2\end{array}\right)$ holds for $1\le k\le n?1$. The case $k=2$ was proved by Haemers, Mohammadian and Tayfeh-Rezaie [Linear Algebra Appl., 2010]. In this paper, we propose the full Brouwer's Laplacian spectrum conjecture and we prove the conjecture holds for $k=2$ which also confirm the conjecture of Guan et al. in 2014.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

On cycles in regular 3-partite tournaments

A $c$-partite tournament is an orientation of a complete $c$-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament $D$ is contained in an $m$-, $(m+1)$- or $(m+2)$-cycle for each $m\in \{3,4,\dots ,\left|V\left(D\right)\right|?2\}$, and this conjecture was proved to be correct for $3\le m\le 7$. In 2012, Xu et al. conjectured that every arc of an $r$-regular 3-partite tournament $D$ with $r\ge 2$ is contained in a $(3k?1)$- or $3k$-cycle for $k=2,3,\dots ,r$. They proved that this conjecture is true for $k=2$. In this paper, we confirm this conjecture for $k=3$, which also implies that Volkmann’s conjecture is correct for $m=7,8$.     

Non-Wieferich primes under the abc conjecture

Assuming the abc conjecture, Silverman proved that, for any given positive integer $a?2$, there are $?\log ?x$ primes $p?x$ such that $a^{p?1}?1(\mathrm{mod}{p}^2)$. In this paper, we show that, for any given integers $a?2$ and $k?2$, there still are $?\log ?x$ primes $p?x$ satisfying $a^{p?1}?1(\mathrm{mod}{p}^2)$ and $p\equiv 1(\mathrm{mod}k)$, under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.     

Stabilization of Boij–Söderberg decompositions of ideal powers

Given an ideal I we investigate the decompositions of Betti diagrams of the graded family of ideals ${\{I^k\}}_k$ formed by taking powers of I. We prove conjectures of Engström from [5] and show that there is a stabilization in the Boij–Söderberg decompositions of $I^k$ for $k>>0$ when I is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for $k>>0$, the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in k. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams.     

Strengthening the Murty–Simon conjecture on diameter 2 critical graphs

A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty–Simon conjecture, states that any diameter-2-critical graph of order $n$ has at most $?n^2∕4?$ edges, with equality if and only if $G$ is a balanced complete bipartite graph. Many partial results about this conjecture have been obtained, in particular it is known to hold for all sufficiently large graphs, for all triangle-free graphs, and for all graphs with a dominating edge. In this paper, we discuss ways in which this conjecture can be strengthened. Extending previous conjectures in this direction, we conjecture that, when we exclude the class of complete bipartite graphs and one particular graph, the maximum number of edges of a diameter-2-critical graph is at most $?{(n?1)}^2∕4?+1$. The family of extremal examples is conjectured to consist of certain twin-expansions of the 5-cycle (with the exception of a set of thirteen special small graphs). Our main result is a step towards our conjecture: we show that the Murty–Simon bound is not tight for non-bipartite diameter-2-critical graphs that have a dominating edge, as they have at most $?n^2∕4??2$ edges. Along the way, we give a shorter proof of the Murty–Simon conjecture for this class of graphs, and stronger bounds for more specific cases. We also characterize diameter-2-critical graphs of order $n$ with maximum degree $n?2$: they form an interesting family of graphs with a dominating edge and $2n?4$ edges.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.     

Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).