Related papers: A Few More Trees the Chromatic Symmetric Function …
It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…
In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated…
DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic…
The Gy\'arf\'as-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an…
We focus on two specific generalizations of the chromatic symmetric function: one involving universal graphs and the other concerning vertex-weighted graphs. In this paper, we introduce a unified generalization that incorporates both…
We study completely independent spanning trees (CIST), \textit{i.e.}, trees that are both edge-disjoint and internally vertex-disjoint, in split graphs. We establish a correspondence between the existence of CIST in a split graph and some…
The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed in \cite{Yoshinaga2015} that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic…
It is a classical problem in algebraic topology asked by Steenrod which graded rings occur as the cohomology ring of a space. In this paper, we define an algebraic version of the graph colouring, span colouring, and observe the relation…
In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and…
An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two…
Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred…
In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem…
We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of…
We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the…
Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs $G$: every…
We examine the measurable chromatic number of distance colorings on the surface of 2-dimensional spheres of varying radii, showing in particular that similar arguments to those used to raise lower bounds in the plane work for all but a…
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that…
Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered…