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تسجيل مستخدم جديدStability for large forbidden subgraphs
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تأليف
Vladimir Nikiforov
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We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.
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Let $n, k, m$ be positive integers with $ngg mgg k$, and let $mathcal{A}$ be the set of graphs $G$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free coloring of $K_n$
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n$ parts of size at most $k$ each such that there is an edge between any two distinct parts. Let $$ f(n,H) = min {k in mathbb N : mbox{there is an $(n,k)$-graph $G$ such that $H otsubseteq G$}} . $$ Barbanera and Ueckerdt observed that $f(n, H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Turan exponent, i.e., ${rm ex}(n, H) = Theta(n^r)$ for some $r$, we show that $Omega (n^{2/r -1}) = f(n, H) = O (n^{2/r-1} log ^{1/r} n)$. We extend this result to all bipartite graphs for which an upper and a lower Turan exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =Theta(n^{1/3})$ for any fixed $t$.
We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.
The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance colo
ring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grunbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbids a bicolored subgraph $H$. When considering the class $mathcal C$ of graphs with no $F$ as an induced subgraph, it is not hard to see that every graph in $mathcal C$ has bounded chromatic number if and only if $F$ is a complete graph of size at most two. We study this phenomena for the class of graphs with no $F$ as a subgraph for $H$-avoiding coloring. We completely characterize all graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded $H$-avoiding chromatic number for a large class of graphs $H$. As a corollary, our main result implies a characterization of graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.
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