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A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2,2,r}$ r ∈ 4,5,6,7,8, $K_{2,3,4}$, $K_{1*4,4}$, $K_{1*4,5}$, $K_{1*5,4}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2,2,r}$ r ∈ 4,5,6,7,8, the others have been proved not to be U3LC...

The hypertree can be defined in many different ways. Katona and Szabó introduced a new, natural definition of hypertrees in uniform hypergraphs and investigated bounds on the number of edges of the hypertrees. They showed that a k-uniform hypertree on n vertices has at most [...] (nk−1) $\left(nk-1\right)$ edges and they conjectured that the upper bound is asymptotically sharp. Recently, Szabó verified that the conjecture holds by recursively constructing an infinite sequence of k-uniform hypertrees and making complicated...