Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On two conjectures about the proper connection number of graphs   A path in an edge-colored graph is called proper if no two consecutive edges
of the path receive the same color. For a connected graph $G$, the proper
connection number $pc(G)$ of $G$ is defined as the minimum number of colors
needed to color its edges so that every pair of distinct vertices of $G$ are
connected by at least one proper path in $G$. In this paper, we consider two
conjectures on the proper connection number of graphs. The first conjecture
states that if $G$ is a noncomplete graph with connectivity $\kappa(G) = 2$ and
minimum degree $\delta(G)\ge 3$, then $pc(G) = 2$, posed by Borozan et al.~in
[Discrete Math. 312(2012), 2550-2560]. We give a family of counterexamples to
disprove this conjecture. However, from a result of Thomassen it follows that
3-edge-connected noncomplete graphs have proper connection number 2. Using this
result, we can prove that if $G$ is a 2-connected noncomplete graph with
$diam(G)=3$, then $pc(G) = 2$, which solves the second conjecture we want to
mention, posed by Li and Magnant in [Theory \& Appl. Graphs 0(1)(2015), Art.2].

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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat