A note on the coprime power graph of groups
-
Mohammed A. Mutar
and
Adel Salim Tayyah
Abstract
Some years ago, Shitov proved that the chromatic number of the power graph of semi-groups is at most countable, thus answering a question raised by Aalipour et al. about whether this statement holds for groups. Later, Dalal and Kumar proved that the chromatic number of the enhanced power graph of a group is also countable. In the line of these recent works, we first generalize the concept of power graphs to coprime power graph, and then we prove the same type of result for this generalization. Furthermore, we state a conjecture about their common relationship as graphs.
(Communicated by Anatolij Dvurečenskij)
References
[1] Aalipour, G.—Akbari, S.—Cameron, P. J.—Nikandish, R.—Shaveisi F.: On the structure of the power graph and the enhanced power graph of a group, Electron. J. Comb. 24(3) (2017), P3.16.10.37236/6497Search in Google Scholar
[2] Afkhami, F. M.—Jafarzadeh, A.—Khashyarmanesh, K.—Mohammadikhah, S.: On cyclic graphs of finite semigroups, Journal of Algebra and Its Applications, 13(7) (2014), Art. 1450035.10.1142/S0219498814500352Search in Google Scholar
[3] Cameron, P. J.: The power graph of a finite group, II. Journal of Group Theory 13(6) (2010), 779-783.10.1515/jgt.2010.023Search in Google Scholar
[4] Cameron, P. J.—Ghosh, S.: The power graph of a finite group. Discrete Math. 311 (2011), 1220-1222.10.1016/j.disc.2010.02.011Search in Google Scholar
[5] Chakrabarty, I.—Ghosh, S.—Sen, M. K.: Undirected power graphs of semigroups, Semigroup Forum 78 (2009), 410-426.10.1007/s00233-008-9132-ySearch in Google Scholar
[6] Dalal S.—Kumar, J.: Chromatic number of the cyclic graph of infinite semigroup, Graphs and Combinatorics 36 (2020), 109-113.10.1007/s00373-019-02120-4Search in Google Scholar
[7] Howie, J. M.: Fundamentals of Semigroup Theory, Oxford University Press, 1995.10.1093/oso/9780198511946.001.0001Search in Google Scholar
[8] Kelarev, A. V.—Quinn, S. J.: A combinatorial property and power graphs of groups, Contrib. General Algebra 12 (2000), 229-235.Search in Google Scholar
[9] Kelarev, A. V.—Quinn, S. J.: Directed graphs and combinatorial properties of semigroups, J. Algebra 251 (2002), 16-26.10.1006/jabr.2001.9128Search in Google Scholar
[10] Kelarev, A. V.—Quinn, S. J.: A combinatorial property and power graphs of semigroups, Comment. Math. Univ. Carolinae 45 (2004), 1-7.Search in Google Scholar
[11] Shitov, Y.: Coloring the power graph of a semigroup, Graphs and Combinatorics 33 (2017), 485-487.10.1007/s00373-017-1773-8Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A note on the coprime power graph of groups
- Simplified axiomatic system of DRl-semigroups
- Special filters in bounded lattices
- Reichenbach’s causal completeness of quantum probability spaces
- A construction of magmas and related representation
- Extensions of the triangular D(3)-Pair {3, 6}
- Hermite-Hadamard type inequalities for new class h-convex mappings utilizing weighted generalized fractional integrals
- Divergence operator of regular mappings
- Monotonicity of the ratio of two arbitrary gaussian hypergeometric functions
- Oscillation and asymptotic criteria for certain third-order neutral differential equations involving distributed deviating arguments
- Multiplicity results for a fourth-order elliptic equation of p(x)-kirchhoff type with weights
- Singular discrete dirac equations
- Convergence of bivariate exponential sampling series in logarithmic weighted spaces of functions
- Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application
- Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations
- On almost cosymplectic generalized (k, μ)ʹ-spaces
- On some recent selective properties involving networks
- Minimal usco and minimal cusco maps and the topology of pointwise convergence
- Corrigendum to: Every positive integer is a sum of at most n + 2 centered n-gonal numbers
Articles in the same Issue
- A note on the coprime power graph of groups
- Simplified axiomatic system of DRl-semigroups
- Special filters in bounded lattices
- Reichenbach’s causal completeness of quantum probability spaces
- A construction of magmas and related representation
- Extensions of the triangular D(3)-Pair {3, 6}
- Hermite-Hadamard type inequalities for new class h-convex mappings utilizing weighted generalized fractional integrals
- Divergence operator of regular mappings
- Monotonicity of the ratio of two arbitrary gaussian hypergeometric functions
- Oscillation and asymptotic criteria for certain third-order neutral differential equations involving distributed deviating arguments
- Multiplicity results for a fourth-order elliptic equation of p(x)-kirchhoff type with weights
- Singular discrete dirac equations
- Convergence of bivariate exponential sampling series in logarithmic weighted spaces of functions
- Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application
- Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations
- On almost cosymplectic generalized (k, μ)ʹ-spaces
- On some recent selective properties involving networks
- Minimal usco and minimal cusco maps and the topology of pointwise convergence
- Corrigendum to: Every positive integer is a sum of at most n + 2 centered n-gonal numbers
