Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.11437

Rainbow connecting $2$-colorings of super-Dirac graphs

János Barát·Simona Boyadzhiyska·Andrea Freschi
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $G$ be a graph with minimum degree $δ(G)\ge|V(G)|/2$. Can we color the edges of $G$ with red and blue so that every pair of non-adjacent vertices is connected by a path consisting of exactly one red edge and one blue edge? We provide an affirmative answer to this question for a class of graphs that are ``close'' to a complete balanced bipartite graph or the disjoint union of two cliques of the same order. Surprisingly, our methods extend to a much broader class of graphs with minimum degree slightly above $|V(G)|/2$. Furthermore, we answer an asymptotic version of this question in full, proving that every graph $G$ satisfying $δ(G)\ge(|V(G)|-1)/2$ has a $2$-edge-coloring such that almost all pairs of vertices are connected by a rainbow path. In addition, we propose a number of related open problems.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

János Barát, Simona Boyadzhiyska, Andrea Freschi. 2026-09-10. Rainbow connecting $2$-colorings of super-Dirac graphs. https://arxiv.org/abs/2609.11437

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗