Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On coloring of graphs of girth 2l + 1 without longer odd holes   A hole is an induced cycle of length at least 4. Let $\l\ge 2$ be a positive
integer, let ${\cal G}_l$ denote the family of graphs which have girth $2\l+1$
and have no holes of odd length at least $2\l+3$, and let $G\in {\cal G}_{\l}$.
For a vertex $u\in V(G)$ and a nonempty set $S\subseteq V(G)$, let $d(u,
S)=\min\{d(u, v):v\in S\}$, and let $L_i(S)=\{u\in V(G) \mbox{ and } d(u,
S)=i\}$ for any integer $i\ge 0$. We show that if $G[S]$ is connected and
$G[L_i(S)]$ is bipartite for each $i\in\{1, \ldots, \lfloor{\l\over
2}\rfloor\}$, then $G[L_i(S)]$ is bipartite for each $i>0$, and consequently
$\chi(G)\le 4$, where $G[S]$ denotes the subgraph induced by $S$. Let
$\theta^-$ be the graph obtained from the Petersen graph by deleting three
vertices which induce a path, let $\theta^+$ be the graph obtained from the
Petersen graph by deleting two adjacent vertices, and let $\theta$ be the graph
obtained from $\theta^+$ by removing an edge incident with two vertices of
degree 3. For a graph $G\in{\cal G}_2$, we show that if $G$ is 3-connected and
has no unstable 3-cutset then $G$ must induce either $\theta$ or $\theta^-$ but
does not induce $\theta^+$. As corollaries, $\chi(G)\le 3$ for every graph $G$
of ${\cal G}_2$ that induces neither $\theta$ nor $\theta^-$, and minimal
non-3-colorable graphs of ${\cal G}_2$ induce no $\theta^+$.

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The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex