# zbMATH — the first resource for mathematics

Edge-domatic number of a graph. (English) Zbl 0537.05049
An edge-dominating set of a graph G is a subset D of the edge set E(G) with the property that for each edge $$e\in E(G)-D$$ there exists at least one edge $$f\in D$$ adjacent to e. The maximum number of classes of partitions of E(G) into edge-dominating sets is called the edge-domatic number of G and is denoted by ed(G). This concept is studied for complete graphs, complete bipartite graphs, cycles and trees. If T is a tree it is found that $$ed(T)=\delta_ e+1,$$ where $$\delta_ e(G)$$ is the minimum degree of a vertex in the line-graph of G, analogous to $$\delta$$ (G), the minimum degree of G. The main result is that for a finite undirected graph G $$\delta(G)\leq ed(G)\leq \delta_ e(G)+1.$$
Reviewer: C.Hoede

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
dominating set; edge-dominating sets; edge-domatic number
Full Text:
##### References:
 [1] Cockayne E. J., Hedetniemi S. T.: Towards a theory of domination in graphs. Networks 7 (1977), 247-261. · Zbl 0384.05051 · doi:10.1002/net.3230070305 [2] Mulder H. M.: The Interval Function of a Graph. Amsterdam 1980. · Zbl 0446.05039 [3] Ore O.: Theory of Graphs. Providence 1962. · Zbl 0105.35401 [4] Zelinka B.: Odd graphs. Arch. Math. Brno · Zbl 0591.05062 · eudml:18168
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.