Given a simple, undirected, and unweighted graph $G$, and an integer $R$, the objective of the \textsc{Minimum Eccentricity Shortest Path (MESP)} is to decide whether there exists an \emph{isometric path} $P$ in $G$ such that the distance from every vertex in the graph to its nearest vertex in $P$ is at most $R$. In this paper, we prove that MESP admits an $O(n^4)$-time algorithm on $K_{2,3}$-minor-free graphs. Our algorithm has a cubic running time when the inputs are restricted to a cactus.
We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq \chi(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open c...
A graph $G$ is said to be $k$-collapsible if $G$ has minimum degree $k$ and every non-null proper induced subgraph of $G$ has minimum degree less than $k.$ In 2018, Bickle conjectured that the minimum number of vertices of degree $k$ in a $k$-collapsible graph of order $n$ with $k\ge 3$ is ${\rm max}\{\lceil 2n/(2k-1)\...
We show that every planar graph with no $t \times t$ grid minor has treewidth at most $4t +4$. This improves on the previously best known bound of $\frac{9}{2}t - \frac{11}{2}$, due to Gu and Tamaki (2012), and is within a factor $2$ of optimal. A key step in the proof is showing the following result, which might be of...
Wouter Cames van Batenburg, Quentin Claus, G. Joret et al.· 0 citations
The spectral Tur\'an type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recentl...
We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...
We prove that every triangle-free graph with minimum degree at least $\frac{n}{3}$ is $4$-colorable and thereby settle a problem of Brandt and Thomass\'e (2005) at the threshold $\frac{n}{3}$. The number four is best possible. For a positive integer-valued function $f(n)=o(n)$, we relate the chromatic number of $f(n)$-...
Jia-Ao Li, Xin-Yuan Li· 0 citations
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