This paper studies degree-bounded vertex partitions, considers analogues for wellknown results on the chromatic number and graph perfection, and presents two algorithms for constructing degree-bounded vertex partitions. The first algorithm minimizes the number of partition classes. The second algorithm minimizes a weighted sum of the partition classes where the weight of a partition class depen...

a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...

The detour index of a connected graph is defined as the sum of detour distances between all unordered pairs of vertices. We determine the n-vertex unicyclic graphs whose vertices on its unique cycle all have degree at least three with the first, the second and the third smallest and largest detour indices respectively for n ≥ 7.

Topological indices are a class of numerical invariants that predict certain physical and chemical properties molecules. Recently, two novel topological indices, named as Sombor index reduced index, were introduced by Gutman, defined where denotes the degree vertex in . In this paper, our aim is to order trees, unicyclic graphs, bicyclic graphs tricyclic with respect index. We determine first f...