External locally-tree-like graphs

Abstract

we deal with set $L{T}_1$ of those locally-tree-like graphs, which have a minimal number of edges according to the number of vertices. Different characteristics of class $L{T}_1$ are found. If $G$ is an arbitrary graph in $L{T}_1$, we find its linear arboricity $(G)$, i.e. the minimal number of vertex disjoint systems chains covering $G$. It is proved that $(G)=[\frac{\mathrm{\Delta }(G)+1}{2}]$.