Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some results on multithreshold graphs   Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with
thresholds $\theta_1 , \ldots, \theta_k$ (strictly increasing) if one can
assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every
pair of vertices $v,w$, we have $vw \in E(G)$ if and only if the inequality
$\theta_i \leq r_v + r_w$ holds for an odd number of indices $i$. When $k=1$ or
$k=2$, the precise choice of thresholds $\theta_1, \ldots, \theta_k$ does not
matter, as a suitable transformation of the ranks transforms a representation
with one choice of thresholds into a representation with any other choice of
thresholds. Jamison asked whether this remained true for $k \geq 3$ or whether
different thresholds define different classes of graphs for such $k$, offering
\$50 for a solution of the problem. Letting $C_t$ for $t > 1$ denote the class
of $3$-threshold graphs with thresholds $-1, 1, t$, we prove that there are
infinitely many distinct classes $C_t$, answering Jamison's question. We also
consider some other problems on multithreshold graphs, some of which remain
open.

15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th