Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $(3,1)^*$-choosability of planar graphs without adjacent short cycles   A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$
of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$
that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at
most $d$ neighbors of $v$ receive color $\pi(v)$. A graph $G$ is said to be
$(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list
assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al.
\cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are
$(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09}
proved that planar graphs without 4- and l-cycles are $(3,1)^*$-choosable,
where $l\in \{8,9\}$.
  There exist planar graphs containing 4-cycles that are not
$(3,1)^*$-choosable (Crown, Crown and Woodall, 1986 \cite{CCW-86}). This partly
explains the fact that in all above known sufficient conditions for the
$(3,1)^*$-choosability of planar graphs the 4-cycles are completely forbidden.
In this paper we allow 4-cycles nonadjacent to relatively short cycles. More
precisely, we prove that every planar graph without 4-cycles adjacent to 3- and
4-cycles is $(3,1)^*$-choosable. This is a common strengthening of all above
mentioned results. Moreover as a consequence we give a partial answer to a
question of Xu and Zhang \cite{XZ-07} and show that every planar graph without
4-cycles is $(3,1)^*$-choosable.

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The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph