Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

Laplacian coefficients of unicyclic graphs with the number of leaves and girth

Let $G$ be a graph of order $n$ and let $\mathcal{L}(G,\lambda)=\sum_{k=0}^n (-1)^{k}c_{k}(G)\lambda^{n-k}$ be the characteristic polynomial of its Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c}, M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves or vertices of degree two, Linear Algebra and its Applications 431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian coefficients in the set $\mathcal{U}_{n,l}$ of all $n$-vertex unicyclic graphs with the number of leaves $l$, we investigate properties of the minimal elements in the partial set $(\mathcal{U}_{n,l}^g, \preceq)$ of the Laplacian coefficients, where $\mathcal{U}_{n,l}^g$ denote the set of $n$-vertex unicyclic graphs with the number of leaves $l$ and girth $g$. These results are used to disprove their conjecture. Moreover, the graphs with minimum Laplacian-like energy in $\mathcal{U}_{n,l}^g$ are also studied.Comment: 19 page, 4figure

On the strength of connectedness of a random hypergraph

Bollob\'{a}s and Thomason (1985) proved that for each $k=k(n) \in [1, n-1]$, with high probability, the random graph process, where edges are added to vertex set $V=[n]$ uniformly at random one after another, is such that the stopping time of having minimal degree $k$ is equal to the stopping time of becoming $k$-(vertex-)connected. We extend this result to the $d$-uniform random hypergraph process, where $k$ and $d$ are fixed. Consequently, for $m=\frac{n}{d}(\ln n +(k-1)\ln \ln n +c)$ and $p=(d-1)! \frac{\ln n + (k-1) \ln \ln n +c}{n^{d-1}}$, the probability that the random hypergraph models $H_d(n, m)$ and $H_d(n, p)$ are $k$-connected tends to $e^{-e^{-c}/(k-1)!}.$Comment: 16 pages, minor revisions from first draf