Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Optimal $\chi$-Bound for ($P_6$, diamond)-Free Graphs   Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it
contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the
path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The
diamond is the graph obtained from $K_4$ by removing an edge. In this paper we
show that every ($P_6$, diamond)-free graph $G$ satisfies $\chi(G)\le
\omega(G)+3$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and
clique number of $G$, respectively. Our bound is attained by the complement of
the famous 27-vertex Schl\"afli graph. Our result unifies previously known
results on the existence of linear $\chi$-binding functions for several graph
classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem
to imperfect ($P_6$, diamond)-free graphs, a careful analysis of the structure
of those graphs, and a computer search that relies on a well-known
characterization of 3-colourable $(P_6,K_3)$-free graphs.

37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat