Narrow Results

As an extension of ${𝔸𝔾}_z(L)$ (the annihilation graph of the commutator poset (lattice) $L$ with respect to an element $z\in L$), we discuss when ${𝔸𝔾}_I(L)$ (the annihilation graph of the commutator poset (lattice) $L$ with respect to an ideal $I\subseteq L$) is a complete bipartite ($r$-partite) graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of $L$ and graph-theoretic properties of its associated graph ${𝔸𝔾}_I(L)$. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by a counterexample that the class of lower sets of a commutator poset $L$ is properly contained in the class of $m$-ideals of $L$ (i.e. multiplicatively absorptive ideals (sets) of $L$ that are defined by commutator operation).

A connected graph $G$ is double-critical if the chromatic number of $G-x-y$ is two less than that of $G$ whenever $x$ and $y$ are two adjacent vertices of $G$. The double-critical graph conjecture states that the complete graphs are the only double-critical graphs. We give a proof of this conjecture for any double-critical graph that contains at least one universal vertex. Using this result we prove the conjecture for all double-critical graphs.