New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

Forcing clique immersions through chromatic number

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t+4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no independent set of size three contains $K_{2\lfloor n∕5\rfloor }$ as an immersion.     

Minimum degree condition for a graph to be knitted

For a positive integer $k$, a graph is $k$-knitted if for each subset $S$ of $k$ vertices, and every partition of $S$ into (disjoint) parts $S_1,\dots ,S_t$ for some $t\ge 1$, one can find disjoint connected subgraphs $C_1,\dots ,C_t$ such that $C_i$ contains $S_i$ for each $i\in \left[t\right]?\{1,2,\dots ,t\}$. In this article, we show that if the minimum degree of an $n$-vertex graph $G$ is at least $n∕2+k∕2?1$ when $n\ge 2k+3$, then $G$ is $k$-knitted. The minimum degree is sharp. As a corollary, we obtain that $k$-contraction-critical graphs are $?\frac{k}{8}?$-connected.     

On color isomorphic subdivisions

Given a graph H and an integer $k?2$, let $f_k(n,H)$ be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph $K_n$ with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that $f_2(n,H_t)=\mathrm{\Omega }(n^{1+\frac{1}{2t?3}})$ where $H_t$ is the 1-subdivision of the complete graph $K_t$. This answers a question of Conlon and Tyomkyn (2021) [4].     

Properties of G-martingales with finite variation and the application to G-Sobolev spaces

As is known, if $B={\left(B_t\right)}_{t\in [0,T]}$ is a $G$-Brownian motion, a process of form ${\int }_0^t{\eta }_{s}d{〈B〉}_s?{\int }_0^{t}2G\left({\eta }_s\right)ds$, $\eta \in M_G^1(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale cannot be form of ${\int }_0^t{\eta }_{s}ds$ or ${\int }_0^t{\gamma }_{s}d{〈B〉}_s$, $\eta ,\gamma \in M_G^1(0,T)$, which implies that the decomposition for generalized $G$-Itô processes is unique: For arbitrary $\zeta \in H_G^1(0,T)$, $\eta \in M_G^1(0,T)$ and non-increasing $G$-martingales $K,L$, if ${\int }_0^t{\zeta }_{s}d{B}_s+{\int }_0^t{\eta }_{s}ds+K_t=L_t,t\in [0,T],$then we have $\eta \equiv 0$, $\zeta \equiv 0$ and$K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).     

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.     

Vertex partitions of (C3,C4,C6)-free planar graphs

A graph is $(k_1,k_2)$-colorable if it admits a vertex partition into a graph with maximum degree at most $k_1$ and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic $p\ge 3$, in particular their standard forms $C:x^4+y^4+z^4+r{x}^2{y}^2+s{y}^2{z}^2+t{z}^2{x}^2=0$. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then $r,s,t$ belong to ${\mathbb{F}}_{p^2}$ and C is maximal or minimal over ${\mathbb{F}}_{p^2}$. Moreover, in this case we will provide a simple criterion in terms of $r,s,t,p$ that tells whether C is maximal (resp. minimal) over ${\mathbb{F}}_{p^2}$.     

A scaling analysis of a star network with logarithmic weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $L$ requests to transmit and is idle, it tries to access the channel at a rate proportional to $log(1+L)$. A stochastic model of such an algorithm is investigated in the case of the star network, in which $J$ nodes can transmit simultaneously, but interfere with a central node 0 in such a way that node 0 cannot transmit while one of the other nodes does. One studies the impact of the log policy on these $J+1$ interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter $N$ being the norm of the initial state. It is shown that the asymptotic fluid behavior of the system is a consequence of the evolution of the state of the network on a specific time scale $(N^t,t\in (0,1))$. The main result is that, on this time scale and under appropriate conditions, the state of a node with index $j\ge 1$ is of the order of $N^{a_j\left(t\right)}$, with $0\le a_j\left(t\right)<1$, where $t?a_j\left(t\right)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.