Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.08703

Critical edge sets in vertex-critical graphs

Ema Skottova·Raphael Steiner
View original source ↗Read source document ↗Export metadata ↓

Abstract

Criticality is a fundamental notion in graph theory that has been studied continually since its introduction in the early 50s by Dirac. A graph is called $k$-vertex-critical ($k$-edge-critical) if it is $k$-chromatic but removing any vertex (edge) lowers the chromatic number to $k-1$. A set of edges in a graph is called critical if its removal reduces the chromatic number of the graph. In 1970, Dirac conjectured a rather strong distinction between the notions of vertex- and edge-criticality, namely that for every $k\ge 4$ there exists a $k$-vertex-critical graph that does not have any critical edges. This conjecture was proved for $k\ge 5$ by Jensen in 2002 and remains open only for $k=4$. A much stronger version of Dirac's conjecture was proposed by Erd\H{o}s in 1985: Let $k\ge 4$ be fixed, and let $f_k(n)$ denote the largest integer such that there exists a $k$-vertex-critical graph of order $n$ in which no set of at most $f_k(n)$ edges is critical. Is it true that $f_k(n)\rightarrow \infty$ for $n\rightarrow \infty$? Strengthening previous partial results, we solve this problem affirmatively for all $k>4$, proving that $$f_k(n)=\Omega(n^{1/3}).$$ This leaves only the case $k=4$ open. We also show that a stronger lower bound of order $\sqrt{n}$ holds along an infinite sequence of numbers $n$. Finally, we provide a first non-trivial upper bound on the functions $f_k$ by proving that $$f_k(n)=O\left(\frac{n}{(\log n)^{\Omega(1)}}\right)$$ for every $k\ge 4$. Our proof of the lower bound on $f_k(n)$ involves an intricate analysis of the structure of proper colorings of a modification of an earlier construction due to Jensen, combined with a gluing operation that creates new vertex-critical graphs without small critical edge sets from given such graphs. The upper bound is obtained using a variant of Szemer\'{e}di's regularity lemma due to Conlon and Fox.

Explore related subjects

math.CO

Keep this discovery

BibTeXRIS

Ema Skottova, Raphael Steiner. 2025-08-12. Critical edge sets in vertex-critical graphs. https://arxiv.org/abs/2508.08703

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

KEEP EXPLORING

Related papers

arXiv2026

Balanced even cycles in signed graphs:Tur\'an bounds, double covers, and parity obstructions

We study Tur\'an problems for balanced even cycles in simple signed graphs, where signed subgraphs are considered up to switching. For every balanced bipartite signed graph, the signed and ordinary Tur\'an numbers differ by at most a factor of two. Our main structural results concern the underlying graphs that admit a signing in which every $2k$-cycle is unbalanced. We characterize these graphs by the absence of an odd dependence among their $2k$-cycle incidence vectors, give a cohomological formulation, and construct subgraph-minimal obstructions of arbitrarily large order. In particular, there is no finite forbidden-subgraph characterization. We also give an exact closed-walk criterion for cycles in double covers and derive a direct signed breadth-first-search upper bound. As applications, we prove \[ \hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2} \] and study the signed hexagon number $R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$. We characterize the underlying graphs counted by $R_6$ and express it as an extremal problem for ordinary $C_6$-free graphs with a prescribed involution. For every sufficiently large $n$, we construct examples with $\Omega(n^{4/3})$ edges, and we give an equivariant construction attaining the coefficient obtained from the F\"uredi--Naor--Verstra\"ete lower bound by double-cover transfer. Finally, we give $n$-vertex $C_{+10}$-free signed graphs with $\Omega(n^{6/5})$ edges and use octagon examples to illustrate the limitations of theta-freeness as a signing criterion.

math.CO↗
arXiv2026

Fractional DP-colorings of $d$-degenerate locally sparse graphs

Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $\chi_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $\chi_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ \chi_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $\chi_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $\chi_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $\Delta$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.

math.CO↗
arXiv2026

Erd\H{o}s-S\'{o}s for digraphs

It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs. Previously, such tight bounds were not known, even just for directed paths. This can be considered as a directed analog of the recently proved Erd\H{o}s-S\'os conjecture. The result was proved by GPT-6 Astra.

math.CO↗