On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Flow number and circular flow number of signed cubic graphs

Let $\mathrm{\Phi }(G,\sigma )$ and ${\mathrm{\Phi }}_c(G,\sigma )$ denote the flow number and the circular flow number of a flow-admissible signed graph $(G,\sigma )$, respectively. It is known that $\mathrm{\Phi }(G)=?{\mathrm{\Phi }}_c(G)?$ for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that $\mathrm{\Phi }(G,\sigma )?{\mathrm{\Phi }}_c(G,\sigma )<1$ holds also for every flow-admissible signed graph $(G,\sigma )$. This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that $\mathrm{\Phi }(G,\sigma )=3$ if and only if ${\mathrm{\Phi }}_c(G,\sigma )=3$ and if $\mathrm{\Phi }(G,\sigma )\in \{4,5\}$, then $4\le {\mathrm{\Phi }}_c(G,\sigma )\le \mathrm{\Phi }(G,\sigma )$. We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area.     

Irredundance trees of diameter 3

A set D of vertices of a graph $G=(V,E)$ is irredundant if each non-isolated vertex of $G[D]$ has a neighbour in $V?D$ that is not adjacent to any other vertex in D. The upper irredundance number $\mathrm{IR}(G)$ is the largest cardinality of an irredundant set of G; an $\mathrm{IR}(G)$-set is an irredundant set of cardinality $\mathrm{IR}(G)$.The IR-graph of G has the $\mathrm{IR}(G)$-sets as vertex set, and sets D and $D^{\prime }$ are adjacent if and only if $D^{\prime }$ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in $D^{\prime }$. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars $S(2n,2n)$, i.e., trees obtained by joining the central vertices of two disjoint stars $K_{1,2n}$.     

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.     

An improvement to the Hilton-Zhao vertex-splitting conjecture

For a simple graph G, denote by n, $\mathrm{\Delta }(G)$, and ${\chi }^{\prime }(G)$ its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let $G^{?}$ be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that $G^{?}$ must be edge-chromatic critical if $\mathrm{\Delta }(G)>n/3$, and they verified this when $\mathrm{\Delta }(G)\ge \frac{n}{2}(\sqrt{7}?1)\approx 0.82n$. In this paper, we prove it for $\mathrm{\Delta }(G)\ge 0.75n$.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.