Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Graceful and Strongly Graceful Permutations   A graceful labelling of a graph G is an injective function f from the set of
vertices of G into the set {0,1,...,|EG|} such that if edge uv is assigned the
label |f(u)-f(v)| then all edge labels have distinct values. A strong graceful
labelling of a tree T with a perfect matching is a graceful labelling of T with
the additional property that the sum of the vertex labels of each odd labelled
edge add up to |ET|. A lobster or a 2-distant tree is a tree T that contains a
path P such that any vertex of T is a distance at most 2 from a vertex of P. In
this paper, we define generalised strongly graceful permutations and discover
two new permutations in addition to the known permutation that is obtained by
replacing each vertex label f(v) by |ET|- f(v). We use these permutations to
prove, by induction, that a lobster with a perfect matching that consists of
the set of end edges of the lobster, is strongly graceful. Further, we show
that there exist strongly graceful labellings that assign the label 0 to four
specific vertices of any tree belonging to this family of lobsters. By using
the technique developed in this paper we will, further, present a tractable way
for proving an equivalent form of Bermond conjecture which states that all
lobsters are graceful. Two out of a total of three cases of the proposed
equivalent form of Bermond conjecture are completed leaving the third case open
for refutation or completion.

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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex