A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

Homomorphisms of planar (m,n)-colored-mixed graphs to planar targets

An $(m,n)$-colored-mixed graph $G=(V,A_1,A_2,\cdots ,A_m,E_1,E_2,\cdots ,E_n)$ is a graph having m colors of arcs and n colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph G to another $(m,n)$-colored-mixed graph H is a morphism $\phi :V(G)\to V(H)$ such that each edge (resp. arc) of G is mapped to an edge (resp. arc) of H of the same color (and orientation). An $(m,n)$-colored-mixed graph T is said to be $P_g^{(m,n)}$-universal if every graph in $P_g^{(m,n)}$ (the planar $(m,n)$-colored-mixed graphs with girth at least g) admits a homomorphism to T.We show that planar $P_g^{(m,n)}$-universal graphs do not exist for $2m+n\ge 3$ (and any value of g) and find a minimal (in the number vertices) planar $P_g^{(m,n)}$-universal graphs in the other cases.     

On the size of special class 1 graphs and (P3;k)-co-critical graphs

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index ${\chi }^{\prime }(G)$ of G is Δ or $\mathrm{\Delta }+1$. A graph G is class 1 if ${\chi }^{\prime }(G)=\mathrm{\Delta }$, and class 2 if ${\chi }^{\prime }(G)=\mathrm{\Delta }+1$; G is Δ-critical if it is connected, class 2 and ${\chi }^{\prime }(G-e)<{\chi }^{\prime }(G)$ for every $e\in E(G)$. A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least $(n(\mathrm{\Delta }-1)+3)/2$ edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, ${\chi }^{\prime }(G+e)={\chi }^{\prime }(G)+1$ for every $e\in E(\bar{G})$. Such graphs have intimate relation to $(P_3;k)$-co-critical graphs, where a non-complete graph G is $(P_3;k)$-co-critical if there exists a k-coloring of $E(G)$ such that G does not contain a monochromatic copy of $P_3$ but every k-coloring of $E(G+e)$ contains a monochromatic copy of $P_3$ for every $e\in E(\bar{G})$. We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all $(P_3;k)$-co-critical graphs. We prove that if G is a $(P_3;k)$-co-critical graph on $n\ge k+2$ vertices, then$e(G)\ge \frac{k}{2}(n-\lceil \frac{k}{2}\rceil -\epsilon )+\left(\begin{array}{c}\lceil k/2\rceil +\epsilon \\ 2\end{array}\right),$ where ε is the remainder of $n-\lceil k/2\rceil$ when divided by 2. This bound is best possible for all $k\ge 1$ and $n\ge \lceil 3k/2\rceil +2$.     

Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space

In this paper, let $(M^n,g,d\mu )$ be n-dimensional noncompact metric measure space which satisfies Poincaré inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation

${△}_{p,f}v+c{v}^{\sigma }=0,$

where $c\le 0,m>n\ge 1,1<p<\frac{m?1+\sqrt{(m?1)(m+3)}}{2},\sigma \le p?1$ are real constants.     

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 $g=4$.     

Spectral strengthening of a theorem on transversal critical graphs

A transversal set of a graph G is a set of vertices incident to all edges of G. The transversal number of G, denoted by $\tau (G)$, is the minimum cardinality of a transversal set of G. A simple graph G with no isolated vertex is called τ-critical if $\tau (G?e)<\tau (G)$ for every edge $e\in E(G)$. For any τ-critical graph G with $\tau (G)=t$, it has been shown that $|V(G)|\le 2t$ by Erd?s and Gallai and that $|E(G)|\le \left(\begin{array}{c}t+1\\ 2\end{array}\right)$ by Erd?s, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to $|V(G)|+|E(G)|\le \left(\begin{array}{c}t+2\\ 2\end{array}\right)$. In this paper, we prove stronger results via spectrum. Let G be a τ-critical graph with $\tau (G)=t$ and $|V(G)|=n$, and let ${\lambda }_1$ denote the largest eigenvalue of the adjacency matrix of G. We show that $n+{\lambda }_1\le 2t+1$ with equality if and only if G is $t{K}_2$, $K_{s+1}\cup (t?s)K_2$, or $C_{2s?1}\cup (t?s)K_2$, where $2\le s\le t$; and in particular, ${\lambda }_1(G)\le t$ with equality if and only if G is $K_{t+1}$. We then apply it to show that for any nonnegative integer r, we have $n(r+\frac{{\lambda }_1}{2})\le \left(\begin{array}{c}t+r+1\\ 2\end{array}\right)$ and characterize all extremal graphs. This implies a pure combinatorial result that $r|V(G)|+|E(G)|\le \left(\begin{array}{c}t+r+1\\ 2\end{array}\right)$, which is stronger than Erd?s-Hajnal-Moon Theorem and Gyárfás-Lehel Theorem. We also have some other generalizations.