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Alexa Gopaulsingh

Publications and source records attributed to Alexa Gopaulsingh.

12 recordsLinked to original sources
arXiv2026

When are Two Subgroups Independent?

Rosenmann and Ventura asked "What is the right definition of dependence of subgroups for general groups?". Here we aim to answer this question. We consider a definition of subgroup independence which is a special case of a category-theoretic one. It is that: Two subgroups of a group are independent if and only if any two endomorphisms, one acting on each subgroup, can be extended to an endomorphism of the group generated by these subgroups. This definition helps to illuminate that the usual condition of almost disjointness of subgroups (two subgroups $A$ and $B$ are almost disjoint if and only if $A \cap B = \{e\}$, where $e$ is the identity element) is not enough to force independence and here we find necessary and (different) sufficient conditions for subgroup independence. The aim of this note is to introduce this general notion of subgroup independence to the group theory community and to pose the open question of its characterisation. We present the partial results known up to this point. Moreover, we use the progress made so far to give a heuristic algorithm that decides subgroup independence for many cases.

math.GR↗
arXiv2025

Automorphism groups and Distinguishing Colorings of Central and Middle Graphs

Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G. We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply this result to obtain new upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) and provide examples showing that these bounds cannot be improved in general. Moreover, we use idempotent commutative Latin squares and a theorem of Galvin on list edge colorings of bipartite graphs to study the total distinguishing chromatic number of central graphs.

math.CO↗
arXiv2025

On Distinguishing Graphs and Cost Number using Automorphism Representations

A distinguishing coloring of a graph is a vertex coloring such that only the identity automorphism of the graph preserves the coloring. A 2-distinguishable graph is a graph which can be distinguished using 2 colors. The cost $\rho(G)$ of a 2-distinguishable graph is the smallest size of a color set of a distinguishing coloring of $G$. The determining number of a graph, $Det(G)$, is the minimum number of nodes, which if fixed by a coloring, would ensure that the coloring distinguishes the entire graph. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) posed an open problem which asks if $\rho(G)$ and $Det(G)$ can be arbitrarily far apart. It is trivial that it cannot be so for the case $Det(G) = 1$ but the answer was unknown for $Det(G) \geq 2$. We solve this problem for the case $Det(G) = 2$. We show that for the case $Det(G) = 2$, that not only is the cost bounded but in fact it takes small values with $\rho(G) = 2, \ 3$ or $4$. In order to establish this, the concept of the automorphism representation of a graph is developed. Graphs having equivalent automorphism representations implies that they have the same distinguishing number (note that just having isomorphic automorphism groups is not enough for this to hold). This prompts a factoring of graphs by which two graphs are distinguishably equivalent iff they have equivalent automorphism representations.

math.CO↗
arXiv2024

Distinguishing chromatic number of middle and subdivision graphs

Let $G$ be a simple finite connected graph of order $n$ greater than or equal to $3$. We obtain the following results: (1). We apply a result of Hamada and Yoshimura from 1976 and some recent results of Alikhani and Soltani (2020) and Kalinowski and Pilsniak (2015) to determine the distinguishing chromatic number of the middle graph $M(G)$ of the graph $G$. In particular, the distinguishing chromatic number $\chi_{D}(M(G))$ of the middle graph $M(G)$ of the graph $G$ is $\Delta(G)+1$ except for four small graphs $C_{4}, K_{4}, C_{6}$, and $K_{3,3}$, and $\Delta(G)+2$ otherwise. (2). In 2016, Kalinowski, Pilsniak, and Wozniak introduced the total distinguishing number $D''(G)$ of $G$. Inspired by a recent result of Mirafzal (2024), we show that the distinguishing number $D(S(G))$ of the subdivision graph $S(G)$ of $G$ is $D''(G)$. Consequently, $D(S(G))$ is at most $\lceil \sqrt{\Delta(G)}\rceil$. (3). We obtain a sharp upper bound for the distinguishing chromatic number of the subdivision graph $S(G)$ of $G$ in terms of the distinguishing number of $G$.

math.CO↗
arXiv2024

Brooks' type theorems for coloring parameters of locally finite graphs and Konig's Lemma

In the past, analogues to Brooks' theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove these theorems are not provable in ZF (i.e. the Zermelo-Fraenkel set theory without the Axiom of Choice (AC)). Moreover, such theorems follow from Konig's Lemma (every infinite locally finite connected graph has a ray-a weak form of AC) in ZF. In ZF, we formulate new conditions for the existence of the distinguishing chromatic number, the distinguishing chromatic index, the total chromatic number, the total distinguishing chromatic number, the odd chromatic number, and the neighbor-distinguishing index in infinite locally finite connected graphs, which are equivalent to Konig's Lemma. In this direction, we strengthen a recent result of Stawiski from 2023. We also figured out the upper bound for list-distinguishing chromatic number for infinite graphs in ZFC (i.e. the Zermelo-Fraenkel set theory with the Axiom of Choice (AC)).

math.CO↗
arXiv2024

Upper bounds for the list-distinguishing chromatic number

We prove analogs of Brooks' Theorem for the list-distinguishing chromatic number of different classes of simple finite connected graphs. Moreover, we determine two upper bounds for the list-distinguishing chromatic number of a graph G in terms of the coloring number of G and the list-chromatic number of G. We also determine the list-distinguishing chromatic number for various families of graphs (for example: the book graphs).

math.CO↗
arXiv2023

Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice

We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to K\H{o}nig Lemma: (a) Any infinite locally finite connected graph G such that the minimum degree of G is greater than k, has a chromatic number for any fixed integer k greater than or equal to 2. (b) Any infinite locally finite connected graph has a chromatic index. (c) Any infinite locally finite connected graph has a distinguishing number. (d) Any infinite locally finite connected graph has a distinguishing index. Our results strengthen some results of Stawiski from a recent paper on the role of the Axiom of Choice in proper and distinguishing colorings since he assumed that the sets of colors can be well-ordered. We also formulate new conditions for the existence of irreducible proper coloring, minimal edge cover, maximal matching, and minimal dominating set in connected bipartite graphs and locally finite connected graphs, which are either equivalent to AC or K\H{o}nig Lemma. Moreover, we show that if the Axiom of Choice for families of 2 element sets holds, then the Shelah--Soifer graph has a minimal dominating set.

math.CO↗
arXiv2023

On Erdos--Dushnik--Miller theorem without AC

In set theory without the Axiom of Choice, we study the possible placement of Erdos-Dushnik-Miller theorem restricted to an uncountable set of vertices in the hierarchy of weak choice forms. We also answer a part of a question raised by Lajos Soukup.

math.LO↗
arXiv2023

Subalgebra Independence

Subobject independence as morphism co-possibility has recently been defined in [2] and studied in the context of algebraic quantum field theory. This notion of independence is handy when it comes to systems coming from physics, but when directly applied to classical algebras, subobject independence is not entirely satisfactory. The sole purpose of this note is to introduce the notion of subalgebra independence, which is a slight variation of subobject independence, yet this modification enables us to connect subalgebra independence to more traditional notions of independence. Apart from drawing connections between subalgebra independence and coproducts and congruences, we mainly illustrate the notion by discussing examples.

math.CT↗
arXiv2016

On a Well-behaved Relational Generalisation of Rough Set Approximations

We examine non-dual relational extensions of rough set approximations and find an extension which satisfies surprisingly many of the usual rough set properties. We then use this definition to give an explanation for an observation made by Samanta and Chakraborty in their recent paper [P. Samanta and M.K. Chakraborty. Interface of rough set systems and modal logics: A survey. Transactions on Rough Sets XIX, pages 114-137, 2015].

cs.AI↗
arXiv2016

Double Successive Rough Set Approximations

We examine double successive approximations on a set, which we denote by $L_2L_1, \ U_2U_1, U_2L_1,$ $L_2U_1$ where $L_1, U_1$ and $L_2, U_2$ are based on generally non-equivalent equivalence relations $E_1$ and $E_2$ respectively, on a finite non-empty set $V.$ We consider the case of these operators being given fully defined on its powerset $\mathscr{P}(V).$ Then, we investigate if we can reconstruct the equivalence relations which they may be based on. Directly related to this, is the question of whether there are unique solutions for a given defined operator and the existence of conditions which may characterise this. We find and prove these characterising conditions that equivalence relation pairs should satisfy in order to generate unique such operators.

cs.LO↗