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Register a new userDense H-free graphs are almost (chi(H)-1)-partite
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Authors
Peter Allen
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By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erd~os-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-1)-partite graph H whose smallest part has t vertices, and any fixed c>0, there exists a constant C such that whenever G is an n-vertex graph with minimum degree at least ((3r-4)/(3r-1)+c)n, either G contains H, or we can delete at most Cn^(2-1/t) edges from G to yield an r-partite graph.
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Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of extremal planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_mathcal{P}}(n,H)=3n-6$ for all $nge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated ErdH{o}s-Stone Theorem. We then completely determine $ex_{_mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $tge2$ and $rge 3$ are integers. However, determining $ex_{_mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.
The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-partite graph. Furedi gave a short proof that one can choose alpha=epsilon. We give a bound for the relationship of alpha and varepsilon which is asymptotically sharp as epsilon to 0.
A graph $G$ is total weight $(k,k)$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k$ real numbers, there is a proper total $L$-weighting, i.e., a mapping $f: V(G) cup E(G) to mathbb{R}$ such that for each $z in V(G) cup E(G)$, $f(z) in L(z)$, and for each edge $uv$ of $G$, $sum_{e in E(u)}f(e)+f(u) e sum_{e in E(v)}f(e) + f(v)$. This paper proves that if $G$ decomposes into complete graphs of odd order, then $G$ is total weight $(1,3)$-choosable. As a consequence, every Eulerian graph $G$ of large order and with minimum degree at least $0.91|V(G)|$ is total weight $(1,3)$-choosable. We also prove that any graph $G$ with minimum degree at least $0.999|V(G)|$ is total weight $(1,4)$-choosable.
The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint Connected Subgraphs for $H$-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for $H$-free graphs as for Disjoint Paths.
A grounded L-graph is the intersection graph of a collection of L shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number $omega$ has chromatic number at most $17omega^4$. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $chi$-bounded. We also survey $chi$-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.
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