Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some results on the structure of multipoles in the study of snarks   Multipoles are the pieces we obtain by cutting some edges of a cubic graph.
As a result of the cut, a multipole $M$ has dangling edges with one free end,
which we call semiedges. Then, every 3-edge-coloring of a multipole induces a
coloring or state of its semiedges, which satisfies the Parity Lemma.
Multipoles have been extensively used in the study of snarks, that is, cubic
graphs which are not 3-edge-colorable. Some results on the states and structure
of the so-called color complete and color closed multipoles are presented. In
particular, we give lower and upper linear bounds on the minimum order of a
color complete multipole, and compute its exact number of states. Given two
multipoles $M_1$ and $M_2$ with the same number of semiedges, we say that $M_1$
is reducible to $M_2$ if the state set of $M_2$ is a non-empty subset of the
state set of $M_1$ and $M_2$ has less vertices than $M_1$. The function $v(m)$
is defined as the maximum number of vertices of an irreducible multipole with
$m$ semiedges. The exact values of $v(m)$ are only known for $m\le 5$. We prove
that tree and cycle multipoles are irreducible and, as a byproduct, that $v(m)$
has a linear lower bound.

16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: nlin