Powers of ideals associated to (C4,2K2)-free graphs

Let G be a $(C_4,2K_2)$-free graph with edge ideal $I(G)?\mathbb{k}[x_1,\dots ,x_n]$. We show that $I{(G)}^s$ has linear resolution for every $s\ge 2$. Also, we show that every power of the vertex cover ideal of G has linear quotients. As a result, we describe the Castelnuovo–Mumford regularity of powers of $I{(G)}^{\vee }$ in terms of the maximum degree of G.     

Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and I is a finite-codimensional ideal of $\mathbb{F}[t_1,\dots ,t_{?}]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$ acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.     

Realization of PBW-deformations of type An quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation ${\mathfrak{B}}_q(\mathfrak{sl}(n+1,\mathbb{C}))$ of type ${\mathbb{A}}_n$ quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type $(1,2,?,n)$. Moreover, we analyze the quantum algebras ${\mathfrak{B}}_q(\mathfrak{g})$ with $\mathfrak{g}$ of type ${\mathbb{D}}_4$, ${\mathbb{B}}_2$ and ${\mathbb{G}}_2$ and give some examples and counterexamples that can be realized by a ladder Ore extension.     

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units $R^{\times }$, we study the algebra $A_R(G,\sigma )$ consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra $A_R(G;\mathrm{\Sigma })$, and we show that it coincides with $A_R(G,{\sigma }^{?1})$. Given any discrete field ${\mathbb{F}}_d$, we prove a graded uniqueness theorem for $A_{{\mathbb{F}}_d}(G,\sigma )$, and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of $A_{{\mathbb{F}}_d}(G,\sigma )$ is equivalent to minimality of G.     

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 the weak 2-coloring number of planar graphs

For a graph $G=(V,E)$, a total ordering L on V, and a vertex $v\in V$, let ${\mathrm{Wcol}}_2[G,L,v]$ be the set of vertices $w\in V$ for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number ${\mathrm{wcol}}_2(G)$ of G is the least k such that there is a total ordering L on V with $|{\mathrm{Wcol}}_2[G,L,v]|\le k$ for all vertices $v\in V$. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.     

Domination versus total domination in claw-free cubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number $\gamma (G)$ of G is the minimum cardinality of a dominating set in G, while the total domination number ${\gamma }_t(G)$ of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then $\frac{{\gamma }_t(G)}{\gamma (G)}\le \frac{12}{7}$, and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then ${\gamma }_t(G)\le \frac{3}{7}n$, and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19].     

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 $\mathrm{\Delta }(G)\ge 0.75n$.