Graphs missing a connected partition
Foster Tom
TL;DR
The paper studies when a graph's chromatic symmetric function $X_G$ fails to be $e$-positive by showing that certain cut-vertex structures force the nonexistence of a connected partition of some type. It develops a framework that translates connected partitions into partial sums of rearrangements of a partition $\lambda$, then employs the Frobenius coin problem and a sequence of lemmas to establish the nonexistence in several cases, with computer-assisted verification for small parameter regimes. The main result shows that a graph with a cut vertex whose deletion yields at least $5$ connected components, or $4$ components each of size at least $2$, is missing a connected partition of some type $\lambda\vdash n$, hence not $e$-positive, strengthening connections to the degree-four vertex conjectures for trees. It also proves that spiders with four legs are not $e$-positive and demonstrates infinite families of trees with a degree-$4$ vertex adjacent to a leaf that still exhibit non-$e$-positivity in light of complete connected-partition realizations. These contributions sharpen the boundary between $e$-positive and non-$e$-positive graphs and push forward the program of characterizing $e$-positivity for trees with high-degree vertices.
Abstract
We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.