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Diane Castonguay

Publications and source records attributed to Diane Castonguay.

16 recordsLinked to original sources
arXiv2024

An infinite family of Type 1 fullerene nanodiscs

A total coloring of a graph colors all its elements, vertices and edges, with no adjacency conflicts. The Total Coloring Conjecture (TCC) is a sixty year old challenge, says that every graph admits a total coloring with at most maximum degree plus two colors, and many graph parameters have been studied in connection with its validity. If a graph admits a total coloring with maximum degree plus one colors, then it is Type 1, whereas it is Type 2, in case it does not admit a total coloring with maximum degree plus one colors but it does satisfy the TCC. Cavicchioli, Murgolo and Ruini proposed in 2003 the hunting for a Type 2 snark with girth at least 5. Brinkmann, Preissmann and Sasaki in 2015 conjectured that there is no Type 2 cubic graph with girth at least 5. We investigate the total coloring of fullerene nanodiscs, a class of cubic planar graphs with girth 5 arising in Chemistry. We prove that the central layer of an arbitrary fullerene nanodisc is 4-total colorable, a necessary condition for the nanodisc to be Type 1. We extend the obtained 4-total coloring to a 4-total coloring of the whole nanodisc, when the radius satisfies r = 5 + 3k, providing an infinite family of Type 1 nanodiscs.

math.CO↗
arXiv2024

Computing a 3-role assignment is polynomial-time solvable on complementary prisms

A $r$-role assignment of a simple graph $G$ is an assignment of $r$ distinct roles to the vertices of $G$, such that two vertices with the same role have the same set of roles assigned to related vertices. Furthermore, a specific $r$-role assignment defines a role graph, in which the vertices are the distinct $r$ roles, and there is an edge between two roles whenever there are two related vertices in the graph $G$ that correspond to these roles. We consider complementary prisms, which are graphs formed from the disjoint union of the graph with its respective complement, adding the edges of a perfect matching between their corresponding vertices. In this work, we characterize the complementary prisms that do not admit a $3$-role assignment. We highlight that all of them are complementary prisms of disconnected bipartite graphs. Moreover, using our findings, we show that the problem of deciding whether a complementary prism has a $3$-role assignment can be solved in polynomial time.

cs.DM↗
arXiv2022

Perfect matching cuts partitioning a graph into complementary subgraphs

In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $Π$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such a way that the edge cut $[V(H),V(\overline{H})]$ satisfies the property $Π$. Motivated by previous work, we consider Comp-Sub($Π$) when the property $Π=\mathcal{PM}$ specifies that the edge cut of the decomposition is a perfect matching. We prove that Comp-Sub($\mathcal{PM}$) is GI-hard when the graph $G$ is $\{C_{k\geq 7}, \overline{C}_{k\geq 7} \}$-free. On the other hand, we show that Comp-Sub($\mathcal{PM}$) is polynomial-time solvable on $hole$-free graphs and on $P_5$-free graphs. Furthermore, we present characterizations of Comp-Sub($\mathcal{PM}$) on chordal, distance-hereditary, and extended $P_4$-laden graphs.

cs.DS↗
arXiv2021

Most direct product of graphs are Type 1

A \textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which the graph $G$ has a $k$-total coloring. Clearly, this number is at least $Δ(G)+1$, where $Δ(G)$ is the maximum degree of $G$. When the lower bound is reached, the graph is said to be Type~1. The upper bound of $Δ(G)+2$ is a central problem that has been open for fifty years, is verified for graphs with maximum degree 4 but not for regular graphs. Most classified direct product of graphs are Type~1. The particular cases of the direct product of cycle graphs $C_m \times C_n$, for $m =3p, 5\ell$ and $8\ell$ with $p \geq 2$ and $\ell \geq 1$, and arbitrary $n \geq 3$, were previously known to be Type 1 and motivated the conjecture that, except for $C_4 \times C_4$, all direct product of cycle graphs $C_m \times C_n$ with $m,n \geq 3$ are Type 1. We give a general pattern proving that all $C_m \times C_n$ are Type 1, except for $C_4 \times C_4$. dditionally, we investigate sufficient conditions to ensure that the direct product reaches the lower bound for the total chromatic number.

math.CO↗
arXiv2019

A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.

cs.DM↗
arXiv2017

Results for the maximum weight planar subgraph problem

The problem of finding the maximum-weight, planar subgraph of a finite, simple graph with nonnegative real edge weights is well known in industrial and electrical engineering, systems biology, sociology and finance. As the problem is known to be NP-hard, much research effort has been devoted over the years to attempt to improve a given approximate solution to the problem by using local moves applied to a planar embedding of the solution. It has long been established that any feasible solution to the problem, a maximal planar graph, can be transformed into any other (having the same vertex set) in a finite sequence of local moves of based on: (i) edge substitution and (ii) vertex relocation and it has been conjectured that moves of only type (i) are sufficient. In this note we settle this conjecture in the affirmative. Furthermore, contrary to recent supposition, we demonstrate that any maximal spanning tree of the original graph is not necessarily a part of any optimal solution to the problem. We hope these results will be useful in the design of future approximate methods for the problem.

cs.DM↗
arXiv2017

(m,n)-Quasitilted and (m,n)-Almost Hereditary Algebras

Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting modules, we introduce (m,n)-almost hereditary algebras. These are the algebras with global dimension m+n and such that any indecomposable module has projective dimension at most m, or else injective dimension at most n. We relate these two classes of algebras, among which (m,1)-almost hereditary ones play a special role. For these, we prove that any indecomposable module lies in the right part of the module category, or else in an m-analog of the left part. This is based on the more general study of algebras the module categories of which admit a torsion-free subcategory such that any indecomposable module lies in that subcategory, or else has injective dimension at most n.

math.RT↗
arXiv2015

Polynomial recognition of cluster algebras of finite type

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky (2006) presented an easier way to verify if a given algebra is of finite type, by testing that all chordless cycles of the graph related to the algebra are cyclically oriented and that there exists a positive quasi-Cartan companion of the skew-symmetrizable matrix related to the algebra. We develop an algorithm that verifies these conditions and decides whether or not a cluster algebra is of finite type in polynomial time. The second part of the algorithm is used to prove that the more general problem to decide if a matrix has a positive quasi-Cartan companion is in NP.

math.AC↗
arXiv2015

A GPU-based parallel algorithm for enumerating all chordless cycles in graphs

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. We propose a GPU parallel algorithm for enumerating all chordless cycles of such a graph. The algorithm, implemented in OpenCL, is based on a previous sequential algorithm developed by the current authors for the same problem. It uses a more compact data structure for solution representation which is suitable for the memory-size limitation of a GPU. Moreover, for graphs with a sufficiently large amount of chordless cycles, the algorithm presents a significant improvement in execution time that outperforms the sequential method.

cs.DC↗
arXiv2015

Polynomial enumeration of chordless cycles on cyclically orientable graphs

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. A graph is called cyclically orientable if it admits an orientation in which every chordless cycle is cyclically oriented. We propose an algorithm to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithm have the advantage of finding each chordless cycle only once in time complexity $\mathcal{O}(n^2)$ in the input size, where $n$ is the number of vertices.

cs.DS↗
arXiv2015

Algorithms and Properties for Positive Symmetrizable Matrices

Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices. Here, symmetrizable matrix are those which are symmetric when multiplied by a diagonal matrix with positive entries called symmetrizer matrix. Four algorithms are developed: one to decide whether there is a symmetrizer matrix; second to find such symmetrizer matrix; another to decide whether the matrix is positive or not; and the last to find a positive quasi-Cartan companion matrix, if there exists. The third algorithm is used to prove that the problem to decide if a matrix has a positive quasi-Cartan companion is NP.

cs.DM↗
arXiv2014

Efficient Enumeration of Chordless Cycles

In a finite undirected simple graph, a {\it chordless cycle} is an induced subgraph which is a cycle. We propose two algorithms to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithms have the advantage of finding each chordless cycle only once. To ensure this, we introduced the concepts of vertex labeling and initial valid vertex triplet. To guarantee that the expansion of a given chordless path will always lead to a chordless cycle, we use a breadth-first search in a subgraph obtained by the elimination of many of the vertices from the original graph. The resulting algorithm has time complexity $\mathcal{O}(n + m)$ in the output size, where $n$ is the number of vertices and $m$ is the number of edges.

cs.DS↗
arXiv2011

Algebras Determined by Their Supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.

math.RT↗
arXiv2010

Toupie algebra, some examples of laura algebras

We consider the bound quiver algebras whose ordinary quiver is that of a canonical algebra. We determine which of those algebras are hereditary, tilted, quasitilted, weakly shod or laura algebras.

math.RT↗
arXiv2005

Fundamental groups and presentations of algebras

In this note, we investigate how different fundamental groups of presentations of a fixed algebra $A$ can be. For finitely many finitely presented groups $G_i$, we construct an algebra $A$ such that all $G_i$ appear as fundamental groups of presentations of $A$.

math.RA↗