Upper-critical graphs

TL;DR

This paper introduces upper-critical graphs, showing they are exactly the complete k-partite graphs, and provides characterizations based on hereditary properties, transformations, and construction methods.

Contribution

It defines the concept of upper-critical graphs and proves their equivalence to complete k-partite graphs, offering new characterizations and insights.

Findings

Upper-critical graphs are exactly the complete k-partite graphs.

Characterizations involve hereditary properties, subgraphs, and minors.

Provides construction and counting methods for these graphs.

Abstract

This work introduces the concept of \emph{upper-critical graphs}, in a complementary way of the conventional (lower)critical graphs: an element $x$ of a graph $G$ is called \emph{critical} if $\chi(G-x)<\chi(G)$. It is said that $G$ is a \emph{critical graph} if every element (vertex or edge) of $G$ is critical. Analogously, a graph $G$ is called \emph{upper-critical} if there is no edge that can be added to $G$ such that $G$ preserves its chromatic number, i.e. \{$e \in E(\bar{G}) \; | \; \chi(G+e) = \chi(G)$ \} $=$ $\emptyset$. We show that the class of upper-critical graphs is the same as the class of complete $k$-partite graphs. A characterization in terms of hereditary properties under some transformations, e.g. subgraphs and minors and in terms of construction and counting is given.