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Register a new userI,F-partitions of Sparse Graphs
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Added by
Derrick Stolee
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than $frac{5}{2}$ has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].
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Szemeredis Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as gentle classes.
An (improper) graph colouring has defect $d$ if each monochromatic subgraph has maximum degree at most $d$, and has clustering $c$ if each monochromatic component has at most $c$ vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than $frac{2d+2}{d+2} k$ is $k$-choosable with defect $d$. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree $m$, no $(1-epsilon)m$ bound on the number of colours was previously known. The above result with $d=1$ solves this problem. It implies that every graph with maximum average degree $m$ is $lfloor{frac{3}{4}m+1}rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree $m$ is $lfloor{frac{7}{10}m+1}rfloor$-choosable with clustering $9$, and is $lfloor{frac{2}{3}m+1}rfloor$-choosable with clustering $O(m)$. As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.
A set of vertices X of a graph G is convex if it contains all vertices on shortest paths between vertices of X. We prove that for fixed p, all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time.
ErdH{o}s, Gyarfas and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Omega(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 log r $ vertex-disjoint monochromatic cycles. This answers a question of Korandi, Mousset, Nenadov, v{S}kori{c} and Sudakov.
By considering graphs as discrete analogues of Riemann surfaces, Baker and Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for graphs. Building on this idea, a concept of gonality for graphs has been defined and has generated much recent interest. We show that there are connected graphs of treewidth 2 of arbitrarily high gonality. We also show that there exist pairs of connected graphs ${G,H}$ such that $Hsubseteq G$ and $H$ has strictly lower gonality than $G$. These results resolve three open problems posed in a recent survey by Norine (Surveys in Combinatorics 2015).
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