On the Independence Number of the Prime-Coprime Graph of a Finite Group

Abstract: The prime-coprime graph $Θ(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all finite groups $G$ for which $Θ(G)$ is a split graph. We establish a general lower bound for the independence number of $Θ(G)$ of an arbitrary finite group $G$. Moreover, we explicitly compute the independence number of $Θ(G)$ for several distinguished families of finite groups, including cyclic, dihedral, dicyclic, and semidihedral groups.

• The paper provides a complete characterization of split and complete-split prime-coprime graphs by linking group element orders with graph independence.
• It establishes lower bounds via semiprime divisors and computes exact independence numbers for classical families such as cyclic, dihedral, dicyclic, and semidihedral groups.
• The results reveal a strong connection between group order structures and combinatorial graph properties, paving the way for further algorithmic and theoretical research.

Independence Number of the Prime-Coprime Graph of Finite Groups

Introduction and Definitions

The prime-coprime graph $\Theta(G)$ of a finite group $G$ is defined with vertex set $G$; two distinct elements $x, y$ are adjacent if and only if the greatest common divisor $\gcd(|x|, |y|)$ is $1$ or a prime. This construction enriches the connections between group structure and graph theory, allowing investigation of algebraic properties via combinatorial invariants.

The independence number $\alpha(\Gamma)$ of a finite simple graph $\Gamma$ is the maximal size of a subset of vertices containing no adjacent pairs. Determining $\alpha(\Theta(G))$ is computationally hard in general, yet provides insight into both the group and graph theoretic properties encoded by $\Theta(G)$.

Characterization of Split and Complete Split Graphs

A principal structural theorem in the paper precisely characterizes when $G$0 is a split graph: $G$1 is split if and only if the subgraph induced by the non-prime-order, non-identity elements $G$2 satisfies one of:

1. $G$3 (i.e., $G$4 has no elements of composite order);
2. $G$5 consists solely of elements of order $G$6, for some fixed prime $G$7 and $G$8;
3. $G$9 consists solely of elements of order $G$0, with $G$1 fixed distinct primes.

An immediate corollary is that every split $G$2 is also a complete split graph: the partition $G$3 forms a clique and a maximal independent set, and every vertex of the clique is adjacent to every vertex of the independent set.

In this case, the independence number is given explicitly: $G$4

General Lower Bounds and Exact Computations

For arbitrary finite groups, a general lower bound is established: $G$5 where $G$6 is the set of semiprime divisors of $G$7, and $G$8 is the set of elements of $G$9 whose order is divisible by $x, y$0.

In the case of cyclic groups $x, y$1, an explicit formula for the size $x, y$2 is given. For $x, y$3 or $x, y$4 with $x, y$5 (where $x, y$6 are the primes in the factorization of $x, y$7): $x, y$8 This yields a lower bound for the independence number for cyclic groups.

For $x, y$9 with all $\gcd(|x|, |y|)$0, the lower bound specializes to

$\gcd(|x|, |y|)$1

Explicit examples demonstrate that, while tight for groups of order with at most three distinct prime divisors, this lower bound is not always attained for higher numbers of prime factors.

Exact Independence Numbers for Classical Families

Cyclic Groups

• For $\gcd(|x|, |y|)$2 ($\gcd(|x|, |y|)$3 a prime, $\gcd(|x|, |y|)$4): $\gcd(|x|, |y|)$5.
• For $\gcd(|x|, |y|)$6 ($\gcd(|x|, |y|)$7 distinct primes): $\gcd(|x|, |y|)$8.
• For $\gcd(|x|, |y|)$9 ($1$0 distinct primes, $1$1): $1$2.
• For $1$3, $1$4: $1$5.
• For $1$6, $1$7: $1$8.

Dihedral Groups

For $1$9, the dihedral group of order $\alpha(\Gamma)$0: $\alpha(\Gamma)$1 This follows from a graph decomposition in which $\alpha(\Gamma)$2 is the join of $\alpha(\Gamma)$3 and a complete subgraph.

Dicyclic Groups

For $\alpha(\Gamma)$4, the dicyclic group of order $\alpha(\Gamma)$5:

• If $\alpha(\Gamma)$6 is odd: $\alpha(\Gamma)$7.
• If $\alpha(\Gamma)$8 with $\alpha(\Gamma)$9 odd: $\Gamma$0.

Semidihedral Groups

For $\Gamma$1, the semidihedral group of order $\Gamma$2 with $\Gamma$3, $\Gamma$4 odd: $\Gamma$5

Implications and Future Directions

The exhaustive characterizations for when $\Gamma$6 is split or complete split connect the independence number with fundamental arithmetical properties of the group order structure. These results, particularly the explicit computations for classical families, suggest refined group invariants detectable through independence number analysis.

The results also show that, while lower bounds based on semiprime orders give strong estimates, the true independence number can sometimes exceed these, especially for cyclic groups with several distinct primes in the order.

Potential directions include:

• Extending these computations to other nonabelian groups with complex element orders;
• Investigating the independence number in connection with other invariants such as clique number, chromatic number, or domination number for $\Gamma$7;
• Studying algorithmic aspects, including efficient approximation methods for $\Gamma$8 in wider classes of finite groups.

Conclusion

This work provides a substantial analytic advancement in understanding the independence number of the prime-coprime graph $\Gamma$9 for finite groups. Complete characterizations for split (and complete split) cases and the explicit computation for several group families establish new benchmarks for the combinatorial analysis of algebraic objects. These results reinforce the deep interaction between group order structure and combinatorial graph properties encoded by $\alpha(\Theta(G))$0 and prompt further study into other graph invariants and group classes.

Reference:

"On the Independence Number of the Prime-Coprime Graph of a Finite Group" (2604.18475)