5 problems

Let $j\ge 1$ and $k\ge 2$ be integers, and let $G$ be a $j$-connected $K_{1,j(k-1)+1}$-free graph. Jackson–Wormald's k-tree conjecture. Then $G$ has a $k$-tree. The paper presents…

Let a k-tree be a graph obtained from a complete graph on $k+1$ vertices by repeatedly adding a vertex adjacent to all vertices of an existing $k$-clique. An odd coloring of a grap…

Let $G$ be the straight linear $K$-tree with $n$ vertices and parameter $k\geq 1$, and let $H$ be the straight linear $K$-tree with $n+1$ vertices. Write $r_G(u,v)$ for the resista…

A $k$-tree is a graph obtained from $K_k$ by iteratively adding a vertex whose neighborhood is a $k$-clique in the existing graph. The join $G\vee H$ is obtained from the disjoint…

Heath and Pemmaraju's conjecture. Every $k$-tree has a $k$-page book embedding.