A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel [ J. Graph Theory, 99 (2022), pp. 217--230] proved that every 1-planar graph with minimum degree 3 and $n\geq 7$ vertices has a matching of size at least $\frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n \geq 36$ vertices has a matching of size at least $\frac{3n+4}{7}$, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel [ J. Graph Theory, 99 (2022), pp. 217--230].

Let [Formula: see text] be a nonseparable plane graph on [Formula: see text] vertices with at least two edges. Suppose that [Formula: see text] has outer face [Formula: see text] and that every 2-vertex-cut of [Formula: see text] contains at least one vertex of [Formula: see text]. Let [Formula: see text] denote the chromatic polynomial of [Formula: see text]. We show that [Formula: see text] for all [Formula: see text]. This result is a corollary of a more general result that [Formula: see text] for all [Formula: see text], where [Formula: see text] is the multivariate Tutte polynomial of [Formula: see text], [Formula: see text], [Formula: see text] for all [Formula: see text] which are not incident to a vertex of [Formula: see text], [Formula: see text] for all [Formula: see text], [Formula: see text] for all other edges [Formula: see text], and [Formula: see text], [Formula: see text] are suitably chosen intervals with [Formula: see text].

For a family $\mathcal{S}$ of graphs, let $\omega(\mathcal{S})$ be the supremum of $t:1

For a graph G of order $n\ge 2$, an ordering $(x_1,x_2,\ldots, x_n)$ of the vertices in G is called a double‐link ordering of G if $x_1x_2\in E(G)$ and $x_i$ has at least two neighbors in $\{x_1,x_2,\ldots,x_{i-1}\}$ for all $i=3,4,\ldots,n$. This paper shows that certain graphs possessing a kind of double‐link ordering have no chromatic zeros in the interval (1,2). This result implies that all graphs with a 2‐tree as a spanning subgraph, certain graphs with a Hamiltonian path, all complete t‐partite graphs, where $t\ge 3$, and all $(v(G)-\Delta(G)+1)$‐connected graphs G have no chromatic zeros in the interval (1,2).