Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proof of a conjecture on the zero forcing number of a graph   Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of
the $k$-forcing number of graph for a positive integer $k$ as the
generalization of the zero forcing number of a graph. The $k$-forcing number of
a simple graph $G$, denoted by $F_k(G)$, is the minimum number of vertices that
need to be initially colored so that all vertices eventually become colored
during the discrete dynamical process by the following rule. Starting from an
initial set of colored vertices and stopping when all vertices are colored: if
a colored vertex has at most $k$ non-colored neighbors, then each of its
non-colored neighbors become colored. Particulary, $F_1(G)$ is a widely studied
invariant with close connection to the maximum nullity of a graph, under the
name of the zero forcing number, denoted by $Z(G)$. Among other things, the
authors proved that for a connected graph $G$ of order $n$ with
$\Delta=\Delta(G)\geq 2$, $Z(G)\leq \frac{(\Delta-2)n+2}{\Delta-1}$, and this
inequality is sharp. Moreover, they conjectured that
$Z(G)=\frac{(\Delta-2)n+2}{\Delta-1}$ if and only if $G=C_n$, $G=K_{\Delta+1}$
or $G=K_{\Delta, \Delta}$. In this note, we show the above conjecture is true.

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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc