Subdivisions of maximal 3-degenerate graphs of order $d+1$ in graphs of minimum degree $d$

Simultaneous coloring of vertices and incidences of graphs

An $n$-subdivision of a graph $G$ is a graph constructed by replacing a path of length $n$ instead of each edge of $G$ and an $m$-power of $G$ is a graph with the same vertices as $G$ and any two vertices of $G$ at distance at most $m$ are adjacent. The graph $G^{\frac{m}{n}}$ is the $m$-power of the $n$-subdivision of $G$. In [M. N. Iradmusa, M. Mozafari-Nia, A note on coloring of $\frac{3}{3}$-power of subquartic graphs, Vol. 79, No.3, 2021] it was conjectured that the chromatic number of $\frac{3}{3}$-power of graphs with maximum degree $Δ\geq 2$ is at most $2Δ+1$. In this paper, we introdu…

非退化线性码的图的顶

Tops of graphs of non-degenerate linear codes

本文研究了有限域上非退化线性码图 $Γ(n,k)_q$ 的极大团，证明了其包含于直线与否时的极大性，并揭示了此类极大团集合的自同构群与 $Γ(n,k+1)_q$ 自同构群的同构关系。

有向图的收缩和细分变体上的 Cops and Robber

Cops and robber on variants of retracts and subdivisions of oriented graphs

研究定向图上 Cops and Robber 博弈的三种变体（强、普通、弱警察模型），证明强和普通警察数在相应收缩变体中保持不变，并证实所有变体在 2-degenerate 二部图上仍具计算难度。

Hat guessing numbers of strongly degenerate graphs

Assume $n$ players are placed on the $n$ vertices of a graph $G$. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a colour out of $q$ available colours. The players can see the hat of each of their neighbours in $G$, but not their own hat. Using a prediscussed guessing strategy, the players then simultaneously guess the colour of their hat. The players win if at least one of them guesses correctly, else the adversary wins. The largest integer $q$ such that there is a winning strategy for the players is denoted by $\text{HG}(G)$, and this…

Clique-coloring of $K_{3,3}$-minor free graphs

A clique-coloring of a given graph $G$ is a coloring of the vertices of $G$ such that no maximal clique of size at least two is monocolored. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a clique-coloring. It has been proved that every planar graph is $3$-clique colorable and every claw-free planar graph, different from an odd cycle, is $2$-clique colorable. In this paper, we generalize these results to $K_{3,3}$-minor free ($K_{3,3}$-subdivision free) graphs.

Degenerate flag varieties: moment graphs and Schröder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal ste…