Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.04401

On $(r,c)$-constant, planar and circulant graphs

Yair Caro·Xandru Mifsud
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper concerns $(r,c)$-constant graphs, which are $r$-regular graphs in which the subgraph induced by the open neighbourhood of every vertex has precisely $c$ edges. The family of $(r,c)$-graphs contains vertex-transitive graphs (and in particular Cayley graphs), graphs with constant link (sometimes called locally isomorphic graphs), $(r,b)$-regular graphs, strongly regular graphs, and much more. This family was recently introduced in [arXiv:2312.08777] serving as important tool in constructing flip graphs [arXiv:2312.08777, arXiv:2401.02315]. In this paper we shall mainly deal with the following: i. Existence and non-existence of $(r, c)$-planar graphs. We completely determine the cases of existence and non-existence of such graphs and supply the smallest order in the case when they exist. ii. We consider the existence of $(r, c)$-circulant graphs. We prove that for $c \equiv 2 \ (\mathrm{mod} \ 3)$ no $(r,c)$-circulant graph exists and that for $c \equiv 0, 1 \ (\mathrm{mod} \ 3)$, $c > 0$ and $r \geq 6 + \sqrt{\frac{8c - 5}{3}}$ there exists $(r,c)$-circulant graphs. Moreover for $c = 0$ and $r \geq 1$, $(r, 0)$-circulants exist. iii. We consider the existence and non-existence of small $(r,c)$-constant graphs, supplying a complete table of the smallest order of graphs we found for $0 \leq c \leq \binom{r}{2}$ and $r \leq 6$. We shall also determine all the cases in this range for which $(r,c)$-constant graphs don't exist. We establish a public database of $(r,c)$-constant graphs for varying $r$, $c$ and order.

Explore related subjects

math.CO

Keep this discovery

BibTeXRIS

Yair Caro, Xandru Mifsud. 2024-03-07. On $(r,c)$-constant, planar and circulant graphs. https://arxiv.org/abs/2403.04401

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↗