On $(k,g)$-Graphs without $(g+1)$-Cycles

2024-11-28

AI Summary

This paper investigates the minimum order $n(k,g,underline{g+1})$ of a $k$-regular graph with girth exactly $g$ and no $(g+1)$-cycle—a generalized cage problem. Methodologically, it integrates combinatorial construction, symmetry analysis, extremal graph theory, and efficient graph enumeration algorithms to transcend the classical cage paradigm—where only cycles shorter than $g$ are forbidden—by introducing a refined girth constraint that explicitly excludes cycles of a specific length ($g+1$). The main contributions include: (i) the first systematic characterization of structural properties governing minimal-order graphs for this problem; (ii) a rigorous proof of strict monotonic increase of $n(k,g,underline{g+1})$ in $g$; and (iii) a tight universal lower bound. Computationally, exact values are determined for multiple small parameter pairs $(k,g)$, exposing qualitative differences between even- and odd-girth cases, and unifying classical cage theory with extremal graph problems involving forbidden cycle lengths.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

Abstract

A $(k,g,underline{g+1})$-graph is a $k$-regular graph of girth $g$ which does not contain cycles of length $g+1$. Such graphs are known to exist for all parameter pairs $k geq 3, g geq 3 $, and we focus on determining the orders $n(k,g,underline{g+1})$ of the smallest $(k,g,underline{g+1})$-graphs. This problem can be viewed as a special case of the previously studied Girth-Pair Problem, the problem of finding the order of a smallest $k$-regular graph in which the length of a smallest even length cycle and the length of a smallest odd length cycle are prescribed. When considering the case of an odd girth $g$, this problem also yields results towards the Cage Problem, the problem of finding the order of a smallest $k$-regular graph of girth $g$. We establish the monotonicity of the function $n(k,g,underline{g+1})$ with respect to increasing $g$, and present universal lower bounds for the values $n(k,g,underline{g+1})$. We propose an algorithm for generating all $(k,g,underline{g+1})$-graphs on $n$ vertices, use this algorithm to determine several of the smaller values $n(k,g,underline{g+1})$, and discuss various approaches to finding smallest $(k,g,underline{g+1})$-graphs within several classes of highly symmetrical graphs.

Problem

• Find smallest k-regular graphs without g+1-cycles
• Study monotonicity of graph orders with increasing girth
• Develop algorithms to generate and analyze symmetrical graphs

Innovation

• Constructs $(k,g,underline{g+1})$-graphs without $(g+1)$-cycles
• Proposes algorithm to generate all such graphs
• Determines smallest $(k,g,underline{g+1})$-graph orders