Searcharxiv⌕ Search

arXiv subjects

Amitayu Banerjee

Publications and source records attributed to Amitayu Banerjee.

18 recordsLinked to original sources
arXiv2026

Adjacent vertex distinguishing total chromatic number of graph products

The adjacent vertex distinguishing (AVD)-total chromatic number $\chi''_{a}(G)$ of a graph $G$ is the least integer $k$ for which $G$ has a proper total coloring $f$ with $k$ colors such that $C_G(u)\neq C_G(v)$ for every edge $uv\in E(G)$, where $C_G(u)=\{f(u)\}\cup\{f(uw):uw\in E(G)\}$. The AVD-total coloring conjecture (AVD-TCC) asserts that $\chi''_{a}(G)\leq \Delta(G)+3$ for every simple graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In this paper, we prove the AVD-TCC for certain classes of graph products, including Cartesian products, lexicographic products, skew products, cover products, comb products, and Indu--Bala products.

math.CO↗
arXiv2026

AVD Total Colorings of Subdivision Graphs, Joins, and Deleted Lexicographic Products

In 2020, Panda, Verma, and Keerti asked whether the central graph of every graph satisfies the AVD-total coloring conjecture. In this paper, we verify the conjecture for central graphs of regular graphs, complete bipartite graphs, graphs that can be expressed as the join of two graphs of the same order, and several other graph classes, thereby providing partial progress towards this open problem. We further determine the AVD-total chromatic number of subdivision graphs of connected graphs and establish new results on the AVD-total coloring of joins of graphs and deleted lexicographic products.

math.CO↗
arXiv2025

Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$. Kaseasbeh and Erfanian (2021) determined the structure of all Cayley graphs on the dihedral group of order $2n$ for subsets $S$ of size at most three. We extend their work by analyzing the structure of such Cayley graphs for subsets $S$ of size at least four. Our main results are as follows: 1. using a classical result of Burnside and Schur, we determine the automorphism groups of Cayley graphs on dihedral groups of order $2p$, where $p$ ranges over infinitely many primes and $S$ consists only of rotations; 2. if $S$ consists of $4 \le 2k < n$ distinct rotations, then the Cayley graph $\mathrm{Cay}(D_{2n},S)$ is the disjoint union of two isomorphic circulant graphs on $n$ vertices, and 3. if $S$ is a generating set of $4\leq k\leq n$ reflections, then the Cayley graph $\mathrm{Cay}(D_{2n},S)$ is bipartite, forming the disjoint union of $k$ perfect matchings.

math.CO↗
arXiv2025

The List-distinguishing chromatic number of graphs containing only small complete bigraphs

In 2006, Collins and Trenk obtained a general sharp upper bound for the distinguishing chromatic number of a connected graph. Inspired by Catlin's combinatorial techniques from 1978, we establish improved upper bounds for classes of connected graphs that have only small complete bigraphs as induced subgraphs. In this framework, we also consider the list-distinguishing chromatic number of such graphs. We apply Menger's theorem to demonstrate applications of our main result for graphs whose constructions are based on Paley graphs, Cayley graphs on Dihedral groups, and circulant Cayley graphs.

math.CO↗
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

Partition models, Permutations of infinite sets without fixed points, and weak forms of AC

We study new relations of the following statements with weak choice principles in ZF and ZFA. 1. There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. 2. If a field has an algebraic closure then it is unique up to isomorphism. 3. For every infinite set X, there exists a permutation of X without fixed points. Moreover, we prove that the principle ``Any infinite locally finite connected graph has a spanning m-bush for any even integer m greater than or equal to 4'' is equivalent to Kőnig's Lemma in ZF. We also study the new status of different weak choice principles in the finite partition model (a type of permutation model) introduced by B.B. Bruce in 2016. Further, we prove that Van Douwen's Choice Principle holds in two recently constructed known permutation models.

math.LO↗
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↗
arXiv2022

Combinatorial Properties and Dependent choice in symmetric extensions based on Lévy Collapse

We work with symmetric extensions based on Lévy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<κ}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $κ$-distributive and $\mathcal{F}$ is $κ$-complete. Further we observe that if $V$ is a model of ZF + $DC_κ$, then $DC_{<κ}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $κ$-strategically closed and $\mathcal{F}$ is $κ$-complete.

math.LO↗
arXiv2020

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ if $\aleph_{\alpha}$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that $\{w < v : \{w,v\} \in E_{G}\}$ is a clique for each $v\in V_{G}$.

math.LO↗
arXiv2020

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_α$, then the set has size $\aleph_α$ for any regular $\aleph_α$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<ω$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.

math.LO↗
arXiv2018

First order logic without equality on relativized semantics

Let $α\geq 2$ be any ordinal. We consider the class $\mathsf{Drs}_α$ of relativized diagonal free set algebras of dimension $α$. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of $\mathsf{Drs}_α$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class $\mathsf{Drs}_α$ corresponds to first order logic, without equality symbol, with $α$-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

math.LO↗