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.

A perfect forest is a spanning forest of a connected graph G, all of whose components are induced subgraphs of G and such that all vertices have odd degree in the forest. A perfect forest can be thought of as a generalization of a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra.We give two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both of our proofs yield polynomial time algorithms for finding a perfect forest in a connected graph of even order.

A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bújtas and Tuza, generalises both classical hypergraph colourings and the more general Voloshin colourings of hypergraphs. In fact, for r-uniform hypergraphs, classical colourings correspond to (2, r)-colourings while an important instance of Voloshin colourings of r-uniform hypergraphs gives (2, r − 1)-colourings. One intriguing aspect of all these colourings, not present in classical colourings, is that H can have gaps in its (α, β)-spectrum, that is, for k 1 < k 2 < k 3 , H would be (α, β)-colourable using k 1 and using k 3 colours, but not using k 2 colours.In an earlier paper, the first two authors introduced, for σ a partition of r, a very versatile type of r-uniform hypergraph which they called σhypergraphs. They showed that, by simple manipulation of the parameters of a σ-hypergraph H, one can obtain families of hypergraphs which have (2, r − 1)-colourings exhibiting various interesting chromatic properties. They also showed that, if the smallest part of σ is at least 2, then H will never have a gap in its (2, r − 1)-spectrum but, quite surprisingly, they found examples where gaps re-appear when α = β = 2.In this paper we extend many of the results of the first two authors to more general (α, β)-colourings, and we study the phenomenon of the disappearanace and re-appearance of gaps and show that it is not just the behaviour of a particular example but we place it within the context of a more general study of constrained colourings of σ-hypergraphs.A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours.This type of colouring was introduced by Bújtas and Tuza in [2] and studied further in [1,3,4,6,5]. For an r-uniform hypergraph, a (2, r)-colouring is just a colouring in the classical sense, while a (2, r − 1)-colouring is what Caro and Lauri called a non-monochromatic-non-rainbow (NMNR) colouring in [7]. This is a special instance of Voloshin colorings of mixed hypergraphs. The references [8,13] are examples of recent works on colourings of mixed hypergraphs, and [10, 12, 15] even give applications. The book [14] and the up-todate website http://http://spectrum.troy.edu/voloshin/publishe.html are recommended sources for literature on all these types of colourings of hypergraphs.For any hypergraph H, an (α, β)-colouring using exactly k colours is called a k-(α, β)-colouring. The lower chromatic number χ α,β is defined as the least number k for which H has a k-(α, β)-colouring. Similarly, the upper chromatic number χ α,β is the largest k for which H has a k-(α, β)-colouring. These parameters are often simply referred to as χ and χ, repectively, when α, β are clear from the context. The (α, β)-spectrum...

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In Ramsey Theory for graphs we are given a graph G and we are required to find the least n 0 such that, for any n ≥ n 0 , any red/blue colouring of the edges of K n gives a subgraph G all of whose edges are blue or all are red. Here we shall be requiring that, for any red/blue colouring of the edges of K n , there must be a copy of G such that its edges are partitioned equally as red or blue (or the sizes of the colour classes differs by one in the case when G has an odd number of edges). This introduces the notion of balanceable graphs and the balance number of G which, if it exists, is the minimum integer bal(n, G) such that, for any red/blue colouring of E(K n ) with more than bal(n, G) edges of either colour, K n will contain a balanced coloured copy of G as described above. The strong balance number sbal(n, G) is analogously defined when G has an odd number of edges, but in this case we require that there are copies of G with both one more red edge and one more blue edge.These parameters were introduced by Caro, Hansberg and Montejano. These authors also introduce the more general omnitonal number ot(n, G) which requires copies of G containing a complete distribution of the number of red and blue edges over E(G).In this paper we shall catalogue bal(n, G), sbal(n, G) and ot(n, G) for all graphs G on at most four edges. We shall be using some of the key results of Caro et al. which we here reproduce in full, as well as some new results which we prove here. For example, we shall prove that the union of two bipartite graphs with the same number of edges is always balanceable.