Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

On the chromatic number of some P5-free graphs

Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, $V(H)$ can be partitioned into A and B such that $H[A]$ is perfect and $\omega (H[B])<\omega (H)$. We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices, respectively. For two disjoint graphs $F_1$ and $F_2$, we use $F_1\cup F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and use $F_1+F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy|x\in V(F_1)\text{ and }y\in V(F_2)\}$. In this paper, we prove that (i) $(P_5,C_5,K_{2,3})$-free graphs are perfectly divisible, (ii) $\chi (G)\le 2{\omega }^2(G)?\omega (G)?3$ if G is $(P_5,K_{2,3})$-free with $\omega (G)\ge 2$, (iii) $\chi (G)\le \frac{3}{2}({\omega }^2(G)?\omega (G))$ if G is $(P_5,K_1+2K_2)$-free, and (iv) $\chi (G)\le 3\omega (G)+11$ if G is $(P_5,K_1+(K_1\cup K_3))$-free.     

Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted $A_n(V)$, for $n\in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n?1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable V-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are V-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.     

On the CLT for rotations and BV functions

Let $x?x+\alpha$ be a rotation on the circle and let φ be a step function. Denote by ${\phi }_n(x)$ the ergodic sums ${\sum }_{j=0}^{n?1}\phi (x+j\alpha )$. For α in a class containing the rotations with bounded partial quotients and under a Diophantine condition on the discontinuities of φ, we show that ${\phi }_n/{6{\phi }_{n}6}_2$ is asymptotically Gaussian for n in a set of density 1.     

A note on Hamilton sequences of extremal Beltrami differentials

Let X be a hyperbolic Riemann surface and let μ be an extremal Beltrami differential on X with ${6\mu 6}_{\infty }\in (0,1)$. It is proved that, if $\{{?}_n\}$ is a Hamilton sequence of μ, then $\{{?}_n\}$ must be a Hamilton sequence of any extremal Beltrami differential ν contained in $[\mu ]$. This result proved a conjecture of the first author of this paper in 1996. This result is also a generalization of two known results.     

Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let $g(f):={\sum }_{j=1}^k{b}_j(z)f(z+c_j)$ be a shift polynomial of f with small meromorphic coefficients $b_j$ in the sense of $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided α, β, $b_0$ are similar small meromorphic functions, we consider zero distribution of $f^n{(g(f))}^s?b_0$, resp. of $g(f)?\alpha f^n?\beta$.     

Symmetrizers for Schur superalgebras

For the Schur superalgebra $S=S(m|n,r)$ over a ground field K of characteristic zero, we define the symmetrizer $T^{\lambda }[i:j]$ of the ordered pairs of tableaux $(T_i,T_j)$ of the shape λ. We show that the K-span $A_{\lambda ,K}$ of all symmetrizers $T^{\lambda }[i:j]$ has a basis consisting of $T^{\lambda }[i:j]$ for $T_i$ and $T_j$ semistandard. In particular, $A_{\lambda ,K}\ne 0$ if and only if λ is an $(m|n)$-hook partition. In this case, the S-superbimodule $A_{\lambda ,K}$ is identified as $D_{\lambda }{?}_K{D}_{\lambda }^o$, where $D_{\lambda }$ and $D_{\lambda }^o$ are left and right irreducible S-supermodules of the highest weight λ.We define modified symmetrizers $T^{\lambda }\{i:j\}$ and show that their $\mathbb{Z}$-span forms a $\mathbb{Z}$-form $A_{\lambda ,\mathbb{Z}}$ of $A_{\lambda ,\mathbb{Q}}$. We show that every modified symmetrizer $T^{\lambda }\{i:j\}$ is a $\mathbb{Z}$-linear combination of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i,T_j$ semistandard. Using modular reduction to a field K of characteristic $p>2$, we obtain that $A_{\lambda ,K}$ has a basis consisting of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i$ and $T_j$ semistandard.