Core Concepts

All cycle-chord graphs are e-positive.

Abstract

The paper establishes the e-positivity of all cycle-chord graphs by using the composition method developed by the author and Zhou. This method is simpler than the (e)-positivity approach used previously for handling cycle-chords with girth at most 4. The author also provides a combinatorial interpretation of the eI-coefficients of the chromatic symmetric functions of cycle-chords.

The key steps are:

- Derive a formula for the chromatic symmetric function of cycle-chord graphs CCab in terms of the chromatic symmetric functions of tadpole graphs.
- Simplify the formula to obtain an explicit expression for the eI-coefficients, showing that they are all non-negative.
- Provide a combinatorial interpretation of the eI-coefficients based on the structure of the compositions I.

As a consequence, the author concludes that all cycle-chord graphs are e-positive.

The paper also poses a conjecture on the e-positivity of all theta graphs, which generalize cycle-chord graphs.

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by David G.L. W... at **arxiv.org** 10-03-2024

Deeper Inquiries

The e-positivity of cycle-chord graphs has several potential applications across various fields of mathematics and computer science. Firstly, in combinatorial optimization, understanding the e-positivity can lead to better algorithms for graph coloring problems, which are fundamental in scheduling, register allocation in compilers, and frequency assignment in wireless networks. The chromatic symmetric functions, which are central to this study, can provide insights into the structure of solutions to these problems.
Secondly, in algebraic combinatorics, the results on e-positivity can be leveraged to explore the properties of symmetric functions and their applications in representation theory. The e-positivity of cycle-chord graphs may also contribute to the study of symmetric function theory, particularly in understanding the relationships between different bases of symmetric functions, such as Schur functions and elementary symmetric functions.
Moreover, in the realm of statistical physics, the chromatic symmetric functions can be related to partition functions of certain statistical models, where e-positivity may imply certain physical properties, such as stability or phase transitions. This connection can open avenues for interdisciplinary research that bridges combinatorial theory and statistical mechanics.

The techniques developed in this paper, particularly the composition method and the combinatorial interpretations of the eI-coefficients, can be extended to prove the e-positivity conjecture for all theta graphs by employing a similar approach to that used for cycle-chord graphs. The key lies in establishing a recursive relationship among the chromatic symmetric functions of theta graphs, akin to the relationships derived for cycle-chords.
By analyzing the structure of theta graphs, which can be viewed as combinations of paths and cycles, one can apply the modular relations and the triple-deletion property to break down the chromatic symmetric functions into simpler components. This decomposition can facilitate the identification of positive coefficients in the e-expansion.
Furthermore, the combinatorial interpretations of the coefficients, as demonstrated in the case of cycle-chords, can be adapted to theta graphs. By defining appropriate parameters and utilizing the properties of compositions, one can derive expressions for the e-coefficients that maintain non-negativity, thereby proving the e-positivity conjecture for all theta graphs.

Yes, there are significant connections between the e-positivity of cycle-chord graphs and other known results on chromatic symmetric functions of graphs. The concept of e-positivity is closely related to the broader study of positivity in symmetric functions, particularly in the context of graph theory. For instance, the e-positivity of cycle-chord graphs builds upon previous results established for other families of graphs, such as paths, cycles, and K-chains, which have been shown to exhibit similar properties.
Moreover, the techniques used to prove the e-positivity of cycle-chord graphs often draw from established results in the literature, such as the work of Stanley on chromatic symmetric functions and the contributions of Gebhard and Sagan regarding (e)-positivity. The modular relations and combinatorial interpretations developed in this paper resonate with earlier findings, suggesting a unified framework for understanding the positivity of chromatic symmetric functions across various graph classes.
Additionally, the results on e-positivity can provide insights into the Schur positivity of graphs, as e-positive graphs are a subset of Schur positive graphs. This relationship highlights the interconnectedness of different positivity concepts in the study of symmetric functions and their applications in combinatorial enumeration and representation theory.

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