The $k$-core of a graph $G$, $C_k\left(G\right)$, is the maximal induced subgraph $H\subseteq G$ such that $\delta \left(G\right)\ge k$, if it exists. For $k>0$, the $k$-shell of a graph $G$ is the subgraph of $G$ induced by the edges contained in the $k$-core and not contained in the $(k+1)$-core. The core number of a vertex is the largest value for $k$ such that $v\in C_k\left(G\right)$, and the maximum core number of a graph, $\widehat{C}\left(G\right)$, is the maximum of the core numbers of the vertices of $G$. A graph $G$ is $k$-monocore if $\widehat{C}\left(G\right)=\delta \left(G\right)=k$. This paper discusses some basic results on the structure of $k$-cores and...

A graph $G$ is called ${(k,d)}^{*}$-choosable if for every list assignment $L$ satisfying $\left|L\right(v\left)\right|=k$ for all $v\in V\left(G\right)$, there is an $L$-coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored with the same color as itself. In this paper, it is proved that every toroidal graph without chordal 7-cycles and adjacent 4-cycles is ${(4,1)}^{*}$-choosable.

As introduced by F. Harary in 1994, a graph $G$ is said to be an $integral$ $sum$ $graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $f\left(u\right)+f\left(v\right)=f\left(w\right)$ for some vertex $w$ in $G$. We prove that every integral sum graph with a saturated vertex, except the complete graph $K_3$, has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be if it is adjacent to every...