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

The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

In 1963, Corr\'adi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k-1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree $\delta(G) \ge 2k-1$ that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.Comment: 7 pages, 2 figures. To appear in Combinatoric

Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture

On \u3cem\u3eK\u3c/em\u3e\u3cem\u3e\u3csub\u3es,t\u3c/sub\u3e\u3c/em\u3e-minors in Graphs with Given Average Degree

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for \u27balanced\u27 H random graphs provide extremal examples and determine the extremal function. Examples of \u27unbalanced graphs\u27 are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and t → ∞. He also found exact values for D(K2,t) for an infinite series of t. In this paper, we confirm the conjecture of Myers and find asymptotically (in s) exact bounds on D(Ks,t ) for a fixed s and large t

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Stability in the Erdős–Gallai Theorems on cycles and paths: Dedicated to the memory of G.N. Kopylov

The Erdős–Gallai Theorem states that for k≥2, every graph of average degree more than k−2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t−3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k−t2)+t(n−k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn−E(Kn−t). In this paper we prove a stability version of the Erdős–Gallai Theorem: we show that for all n≥3t>3, and k∈{2t+1,2t+2}, every n-vertex 2-connected graph G with e(G)>h(n,k,t−1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t−1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n−t−O(1). © 2016 Elsevier Inc