arXiv · 0707.4499
We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result.
Explore connections, maps & timelines
Vladimir Nikiforov. 2007-11-22. A spectral condition for odd cycles in graphs. https://arxiv.org/abs/0707.4499
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
We introduce the \emph{orbital bivariate chromatic polynomial}, which counts generalized vertex colorings of a graph up to the action of a group of symmetries. More precisely, adjacent vertices must either receive distinct colors from a set of $λ$ colors or may receive the same color from a distinguished subset of $λ-μ$ colors. This new graph polynomial simultaneously generalizes the orbital chromatic polynomial due to Cameron and Kayibi (2007) and bivariate chromatic polynomial due to Dohmen, Pönitz, and Tittmann (2003). We establish fundamental structural properties and derive explicit formulas for several graph families, including complete graphs, complete bipartite graphs, paths, cycles, and wheels. In particular, several of these results are new even in the special case of the orbital chromatic polynomial. For cycles, our results generalize Moreau's necklace polynomial and lead to a new generalization and proof of Fermat's Little Theorem.
We prove that a smooth $d$-manifold $M$ is diffeomorphic to $\mathbb R^d$ if it admits a Lyapunov-Reeb function, i.e., a smooth map $f\colon M\to\mathbb R$ that is proper, lower-bounded, and possesses exactly one critical point. We further establish that the moduli spaces of simple polylines and positively oriented polygons with fixed or varying side lengths are diffeomorphic to Euclidean spaces, by explicitly constructing Lyapunov-Reeb functions on these spaces. This yields a refined resolution to the Carpenter's Rule Problem and settles a conjecture proposed by González and Sedano-Mendoza. In addition, we describe foliation structures of the moduli spaces and develop algorithms for the refolding problem.
We study the extremal problem that relates the spectral radius $λ(G)$ of an $F$-free graph $G$ to its number of edges. Firstly, we prove that for any graph $F$ with chromatic number $χ(F)=r+1\ge 3$, if $G$ is an $F$-free graph with $m$ edges, then $λ^2(G)\le {(1-\frac{1}{r} + o(1))2m}$. This provides a unified extension of both the Erdős-Stone-Simonovits theorem and its vertex-spectral extension due to Nikiforov, and confirms a conjecture proposed by Li, Liu and Feng. We also establish the corresponding edge-spectral stability, showing that if $G$ is an $F$-free graph with $m$ edges with $λ^2(G) \ge (1- \frac{1}{r} - o(1))2m$, then $G$ differs from a complete bipartite graph by $o(m)$ edges when $r=2$, and $G$ differs from an $r$-partite Turán graph by $o(m)$ edges when $r\ge 3$. This extends both the classical Erdős-Simonovits stability and the Nikiforov vertex-spectral stability. This edge-spectral stability theorem is the key ingredient in our companion work on the Brualdi-Hoffman-Turán problem for color-critical forbidden graphs. As an application of our method, we improve a result of Zhai, Lin and Shu by showing that if $λ(G)>\sqrt{m}$, then there exist two vertices in $G$ that have at least $\frac{1}{2}\sqrt{m} - O(1)$ common neighbors. The constant $\frac{1}{2}$ is best possible as witnessed by a random construction.