Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Universality and Extremality of graphs with a distance
  constrained colouring   A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of
non-negative integers (with minimum assignment $0$) to its vertices such that
the adjacent vertices must receive integers at least two apart and vertices at
distance two must receive distinct integers. The lambda chromatic number (or
the $\lambda$ number) of a graph $G$ is the least positive integer among all
the maximum assigned positive integer over all possible lambda colouring of the
graph $G$. Here we have primarily shown that every graph with lambda chromatic
number $t$ can be embedded in a graph, with lambda chromatic number $t$, which
admits a partition of the vertex set into colour classes of equal size. It is
further proved that if an $n-$vertex graph with lambda chromatic number
$t\geq5$, where $n\geq t+1$, contains maximum number of edges, then the vertex
set of such graph admits an equitable partition. For such an admitted equitable
partition there are either $0$ or $\min\{|A|,|B|\}$ number of edges between
each pair $(A,B)$ of subsets (i.e. roughly, such partition is a "sparse like"
equitable partition). Here we establish a classification result, identifying
all possible $n-$vertex graphs with lambda chromatic number $t\geq3$, where
$n\geq t+1$, which contain maximum number of edges. Such classification
provides a solution of a problem posed more than two decades ago by John P.
Georges and David W. Mauro.

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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn