arXiv subjects

Monica M. Y. Wang

Publications and source records attributed to Monica M. Y. Wang.

5 recordsLinked to original sources

arXiv2026

The spiders $S(4m+2,\,2m,\,1)$ are $e$-positive

Using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, as conjectured by Aliniaeifard, van Willigenburg and Wang. Following a divide-and-conquer strategy, we group one or two positive-coefficient $e_J$-terms with each negative-coefficient $e_I$-term, where each $J$ is a composition obtained by rearranging the parts of the composition $I$, and prove that the resulting total coefficient is nonnegative. Our main contribution is the explicit construction of the required injections.

math.CO↗

arXiv2021

The $e$-positivity and Schur positivity of the chromatic symmetric functions of some trees

We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.

math.CO↗

arXiv2021

Two cycle-chord graphs are $e$-positive

We prove Gebhard and Sagan's $(e)$-positivity of the line graphs of tadpoles in noncommuting variables. This implies the $e$-positivity of these line graphs. We then extend this $(e)$-positivity result to that of certain cycle-chord graphs, and derive the bivariate generating function of all cycle-chord graphs.

math.CO↗

arXiv2020

The edge metric dimension of the generalized Petersen graph $P(n,3)$ is 4

It is known that the problem of computing the edge dimension of a graph is NP-hard, and that the edge dimension of any generalized Petersen graph $P(n,k)$ is at least 3. We prove that the graph $P(n,3)$ has edge dimension 4 for $n\ge 11$, by showing semi-combinatorially the nonexistence of an edge resolving set of order 3 and by constructing explicitly an edge resolving set of order 4.

math.CO↗

arXiv2020

Non-Schur-positivity of chromatic symmetric functions

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs with $m=2n-1$.

math.CO↗