Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Erd\H{o}s-Lov\'asz Tihany Conjecture for graphs with forbidden holes   A hole in a graph is an induced cycle of length at least $4$. Let $s\ge2$ and
$t\ge2$ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be
partitioned into two sets $S$ and $T$ such that $\chi(G[S ]) \ge s$ and
$\chi(G[T ]) \ge t$. The well-known Erd\H{o}s-Lov\'asz Tihany Conjecture from
1968 states that every graph $G$ with $\omega(G) < \chi(G) = s + t - 1$ is
$(s,t)$-splittable. This conjecture is hard, and few related results are known.
However, it has been verified to be true for line graphs, quasi-line graphs,
and graphs with independence number $2$. In this paper, we establish more
evidence for the Erd\H{o}s-Lov\'asz Tihany Conjecture by showing that every
graph $G$ with $\alpha(G)\ge3$, $\omega(G) < \chi(G) = s + t - 1$, and no hole
of length between $4$ and $2\alpha(G)-1$ is $(s,t)$-splittable, where
$\alpha(G)$ denotes the independence number of a graph $G$.

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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cs
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin