Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On zero-sum spanning trees and zero-sum connectivity   We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of
a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is
said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)}
f(e) =0$. We study the following type of questions, in several cases obtaining
best possible results: Under which conditions on $|f(G)|$ can we guarantee the
existence of a zero-sum spanning tree of $G$? The types of $G$ we consider are
complete graphs, $K_3$-free graphs, $d$-trees, and maximal planar graphs. We
also answer the question of when any such colouring contains a zero-sum
spanning path or a zero-sum spanning tree of diameter at most $3$, showing in
passing that the diameter-$3$ condition is best possible. Finally, we give, for
$G = K_n$, a sharp bound on $|f(K_n)|$ by which an interesting zero-sum
connectivity property is forced, namely that any two vertices are joined by a
zero-sum path of length at most $4$.
  One feature of this paper is the proof of an Interpolation Lemma leading to a
Master Theorem from which many of the above results follow and which can be of
independent interest.

11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org