3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices

Let $g(k,t)$ be the minimum integer such that every plane graph with girth g at least $g(k,t)$, minimum degree $\delta =2$ and no $(k+1)$-paths consisting of vertices of degree 2, where $k\ge 1$, has a 3-vertex with at least t neighbors of degree 2, where $1\le t\le 3$.In 2015, Jendrol' and Maceková proved $g(1,1)\le 7$. Later on, Hudák et al. established $g(1,3)=10$, Jendrol', Maceková, Montassier, and Soták proved $g(1,1)\ge 7$, $g(1,2)=8$ and $g(2,2)\ge 11$, and we recently proved that $g(2,2)=11$ and $g(2,3)=14$.Thus $g(k,t)$ is already known for $k=1$ and all t. In this paper, we prove that $g(k,1)=3k+4$, $g(k,2)=3k+5$, and $g(k,3)=3k+8$ whenever $k\ge 2$.     

Classification of homogeneous holomorphic two-spheres in complex Grassmann manifolds

In this paper we completely classify the linearly full homogeneous holomorphic two-spheres in the complex Grassmann manifolds $G(2,N)$ and $G(3,N)$. We also obtain the Gauss equation for the holomorphic immersions from a Riemann surface into $G(k,N)$. By using which, we give explicit expressions of the Gaussian curvature and the square of the length of the second fundamental form of these homogeneous holomorphic two-spheres in $G(2,N)$ and $G(3,N)$.     

Graphs with nullity 2c(G)+p(G)−1

Let G be a connected graph with $n(G)$ vertices and $e(G)$ edges. The nullity of G, denoted by $\eta (G)$, is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that $\eta (G)\le 2c(G)+p(G)?1$ unless G is a cycle of order a multiple of 4, where $c(G)=e(G)?n(G)+1$ is the elementary cyclic number of G and $p(G)$ is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity $2c(G)?1$, thus leaving the problem to characterize connected graphs G with nullity $2c(G)+p(G)?1$ when $p(G)\ne 0$. In this paper, we solve this problem completely.     

The space of lines in cyclic covers of projective space

Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ${\mathbb{P}}^n$ ramified over a general hypersurface $X\subset P^n$ of degree md. In this paper we study the space $F(Y)$ of lines in Y and show that it is smooth of dimension $2(n-1)-d(m-1)$ if $md>2n-3$ and $2(n-1)-d(m-1)\ge 0$. When $2(n-1)=d(m-1)$, our result gives a formula on the number of m-contact order lines of X (see Definition 1.2).     

SLE intersecting with random hulls

Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that $\mathrm{SLE}(8/3)$ does not intersect a deterministic hull. For general $\mathrm{SLE}(\kappa )$ with $\kappa \ne 8/3$, no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that $\mathrm{SLE}(\kappa )$ does not intersect this random hull for any $\kappa \in (0,8)$. As a corollary, we will give a new proof of Werner's result on conformal restriction measures.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

Large induced subgraphs with three repeated degrees

For any positive integer k, let $C(k)$ denote the least integer such that any n-vertex graph has an induced subgraph with at least $n?C(k)$ vertices, in which at least $\min ?\{k,n?C(k)\}$ vertices are of the same degree. Caro, Shapira and Yuster initially studied this parameter and showed that $\mathrm{\Omega }(k\log ?k)\le C(k)\le {(8k)}^k$. For the first nontrivial case, the authors proved that $3\le C(3)\le 6$, and the exact value was left as an open problem. In this paper, we first show that $3\le C(3)\le 4$, improving the former result as well as a recent result of Kogan. For special families of graphs, we prove that $C(3)=3$ for $K_5$-free graphs, and $C(3)=1$ for large $C_{2s+1}$-free graphs. In addition, extending a result of Erd?s, Fajtlowicz and Staton, we assert that every $K_r$-free graph is an induced subgraph of a $K_r$-free graph in which no degree occurs more than three times.     

Sobolev homeomorphisms are dense in volume preserving automorphisms

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-$(1,p)$ volume preserving homeomorphisms on closed and connected n-dimensional manifolds, $n\ge 3$, for $p<n?1$. We also prove that if $p>n$ this result is not true. More precisely, we obtain the density of Sobolev-$(1,p)$ homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-$(1,p)$ ball with the same radius centered at the identity.     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

Path decompositions of triangle-free graphs

Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then $p(G)\le ?n/2?$. If G is allowed to be disconnected, then the upper bound $?\frac{3}{4}n?$ for $p(G)$ was obtained by Donald [7], which was improved to $?\frac{2}{3}n?$ independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, $?\frac{2}{3}n?$ is reached and so this bound is tight. If triangles are forbidden in G, then $p(G)\le ?\frac{g+1}{2g}n?$ can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that $p(G)\le ?3n/5?$, which improves the above result with $g=4$.