The stable set problem in graphs with bounded genus and bounded odd cycle packing number

Induced odd cycle packing number, independent sets, and chromatic number

The induced odd cycle packing number $iocp(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al.~\cite{indoc} proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex g…

Min-Max Theorems for Packing and Covering Odd $(u,v)$-trails

We investigate the problem of packing and covering odd $(u,v)$-trails in a graph. A $(u,v)$-trail is a $(u,v)$-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let $ν(u,v)$ denote the maximum number of edge-disjoint odd $(u,v)$-trails, and $τ(u,v)$ denote the minimum size of an edge-set that intersects every odd $(u,v)$-trail. We prove that $τ(u,v)\leq 2ν(u,v)+1$. Our result is tight---there are examples showing that $τ(u,v)=2ν(u,v)+1$---and substantially improves upon the bound of $8$ obtained in [Churchley e…

奇环填充树宽：关于奇次要图类中的最大独立集问题

Odd-Cycle-Packing-treewidth: On the Maximum Independent Set problem in odd-minor-free graph classes

本文提出奇环填充树宽（OCP-tw）参数，证明其奇次要单调性与类似网格定理的结构结果，推导出近似算法，并表明最大独立集问题在OCP-tw有界图上可多项式求解，还将该概念扩展到整数规划矩阵。

Packing cycles in planar and bounded-genus graphs

We devise constant-factor approximation algorithms for finding as many disjoint cycles as possible from a certain family of cycles in a given planar or bounded-genus graph. Here disjoint can mean vertex-disjoint or edge-disjoint, and the graph can be undirected or directed. The family of cycles under consideration must satisfy two properties: it must be uncrossable and allow for an oracle access that finds a weight-minimal cycle in that family for given nonnegative edge weights or (in planar graphs) the union of all remaining cycles in that family after deleting a given subset of edges. Our se…

Cover time of graphs with bounded genus

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all vertices of the graph. It is known that the cover time of any finite connected $n$-vertex graph is at least $(1 + o(1)) n \log n$ and at most $(1 + o(1)) \frac{4}{27} n^3$. By Jonasson and Schramm, the cover time of any bounded-degree finite connected $n$-vertex planar graph is at least $c n(\log n)^2$ and at most $6n^2$, where $c$ is a positive constant depending only on the maximal degree of the graph. In particular, the lower bound is established via the use o…

On S-packing edge-colorings of cubic graphs

Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 ,X 2,. .. ,X k } such that for each 1 $\le$ i $\le$ k, the distance between two distinct edges e, e ' $\in$ X i is at least s i + 1. This paper studies S-packing edge-colorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1,1,1,3,3)-packing edge-colorable, (1,1,1,4,4,4,4,4)-packing edge-colorable and (1,1,2,2,2,2,2)-packing edge-colorable. We determine sharper results for cubic gr…