Planar Induced Subgraphs of Sparse Graphs

We show that every graph has an induced pseudoforest of at least $n-m/4.5$ vertices, an induced partial 2-tree of at least $n-m/5$ vertices, and an induced planar subgraph of at least $n-m/5.2174$ vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest $K_h$-minor-free graph in a given graph can sometimes be at most $n-m/6+o(m)$.Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n