Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Colorings with Rainbow Paths   Let $G$ be a connected graph of chromatic number $k$. For a $k$-coloring $f$
of $G$, a full $f$-rainbow path is a path of order $k$ in $G$ whose vertices
are all colored differently by $f$.
  We show that $G$ has a $k$-coloring $f$ such that every vertex of $G$ lies on
a full $f$-rainbow path, which provides a positive answer to a question posed
by Lin (Simple proofs of results on paths representing all colors in proper
vertex-colorings, Graphs Combin. 23 (2007) 201-203). Furthermore, we show that
if $G$ has a cycle of length $k$, then $G$ has a $k$-coloring $f$ such that,
for every vertex $u$ of $G$, some full $f$-rainbow path begins at $u$, which
solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic
graphs, arXiv 1503.00965v1). Finally, we establish some more results on the
existence of optimal colorings with (directed) full rainbow paths.

19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics