Syed Mujtaba Hassan | Searcharxiv

arXiv2025

On coloring graphs with well-distributed edge density

In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as regular graphs, trees and other popular classes of graphs. The paper aims to explore some interesting properties regarding colorings average hereditary graphs. We prove a new upper bound for the chromatic number of a graph in terms of its maximum average degree and show that this bound is an improvement on previous bounds. From this, we show a relationship between the average degree and the chromatic number of an average hereditary graph. We then show that even with new bound, the graph 3-coloring problem remains NP-hard when the input is restricted to average hereditary graphs. We provide an equivalent condition for a graph to be average hereditary, through which we show that we can decide if a given graph is average hereditary in polynomial time.

cs.DM↗

arXiv2025

Reducibility among NP-Hard graph problems and boundary classes

Many NP-hard graph problems become easy for some classes of graphs. For example, coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum substructure for which the problem remains hard? We use the notion of boundary classes to study such questions. In this paper, we introduce a method for transforming the boundary class of one NP-hard graph problem into a boundary class for another problem. If Π and Γ are two NP-hard graph problems where Π is reducible to Γ, we transform a boundary class of Π into a boundary class of Γ. More formally if Π is reducible to Γ, where the reduction satisfies certain conditions, then X is a boundary class of Π if and only if the image of X under the reduction is a boundary class of Γ. This gives us a relationship between boundary classes and reducibility among several NP-hard problems. To show the strength of our main result, we apply our theorem to obtain some previously unknown boundary classes for a few graph problems namely; vertex-cover, clique, traveling-salesperson, bounded-degree-spanning-tree, subgraph-isomorphism and clique-cover.

cs.CC↗