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Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1, . . . , ±(k − 1)}.It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we give an affirmative answer to this conjecture, except for r = 5.

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Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.

No abstract