Journal of Combinatorial Theory Series B最新文献

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs H of a graph G of the neighborhood set system of H is sandwiched between $\Omega (\log \mathrm{deg}(G\left)\right)$ and $O\left(\mathrm{deg}\right(G\left)\right)$, where $\mathrm{deg}\left(G\right)$ denotes the degeneracy of G. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes.

Then we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy.

Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy.

As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute ε-approximations of size $O(1/\epsilon )$ for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.

A graph G is q-Ramsey for a q-tuple of graphs $(H_1,\dots ,H_q)$ if for every q-coloring of the edges of G there exists a monochromatic copy of $H_i$ in color i for some $i\in \left[q\right]$. Over the last few decades, researchers have investigated a number of questions related to this notion, aiming to understand the properties of graphs that are q-Ramsey for a fixed tuple. Among the tools developed while studying questions of this type are gadget graphs, called signal senders and determiners, which have proven invaluable for building Ramsey graphs with certain properties. However, until now these gadgets have been shown to exist and used mainly in the two-color setting or in the symmetric multicolor setting, and our knowledge about their existence for multicolor asymmetric tuples is extremely limited. In this paper, we construct such gadgets for any tuple of cliques. We then use these gadgets to generalize three classical theorems in this area to the asymmetric multicolor setting.

Given a tree T of order n, one can contract any edge and obtain a new tree $T^{⁎}$ of order $n-1$. In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least $\frac{1}{3}$ in contracting an edge of a tree. In 2023, Luo, Xu, Wagner and Wang proved the case when the edge to be contracted is a pendant edge. In this article, we prove that the conjecture is true in general.

A well-known theorem of Sabidussi shows that a simple G-arc-transitive graph can be represented as a coset graph for the group G. This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely graphs with finite valency and finite edge-multiplicity. The construction yields a G-arc-transitive coset graph $\mathrm{Cos}(G,H,J)$, where $H,J$ are stabilisers in G of a vertex and incident edge, respectively. A first major application is presented concerning arc-transitive maps on surfaces: given a group $G=〈a,z〉$ with $\left|z\right|=2$ and $\left|a\right|$ finite, the coset graph $\mathrm{Cos}(G,〈a〉,〈z〉)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a G-arc-transitive map $(V,E,F)$ (with $V,E,F$ the sets of vertices, edges and faces, respectively), namely, a G-rotary map if $\left|az\right|$ is finite, and a G-bi-rotary map if $\left|z{z}^a\right|$ is finite. The G-rotary map can be represented as a coset geometry for G, extending the notion of a coset graph. However the G-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. In addition a coset geometry construction is given of a flag-regular map $(V,E,F)$ for non necessarily simple graphs. For all of these constructions it is proved that the face boundary cycles are simple cycles precisely when the given group acts faithfully on $V\cup F$. Illustrative examples are given for graphs related to the n-dimensional hypercubes and the Petersen graph.

Let T be an oriented tree on n vertices with maximum degree at most $e^{o\left(\sqrt{\log n}\right)}$. If G is a digraph on n vertices with minimum semidegree ${\delta }^0\left(G\right)\ge (\frac{1}{2}+o(1\left)\right)n$, then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree $o(n/\log n)$). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least $|\mathrm{Aut}(T){|}^{-1}{(\frac{1}{2}-o(1\left)\right)}^{n}n!$ copies of T, which is optimal. This implies the analogous result in the undirected case.

The burning number $b\left(G\right)$ of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that $b\left(G\right)\le \lceil \sqrt{n}\rceil$ for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is $b\left(G\right)\le (1+o(1\left)\right)\sqrt{n}$.

For a finite abelian group A, define $f\left(A\right)$ to be the minimum integer such that for every complete digraph Γ on f vertices and every map $w:E\left(\Gamma \right)\to A$, there exists a directed cycle C in Γ such that ${\sum }_{e\in E\left(C\right)}w\left(e\right)=0$. The study of $f\left(A\right)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f\left(Z_p^k\right)=O\left(pk{(\log k)}^2\right)$, where p is prime, with an improved bound of $O(k\log k)$ when $p=2$. These bounds are tight up to a factor which is polylogarithmic in k.

The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least $k\ge 2$ contains a cycle of length at least $k+1$. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of $k,r$ and n with $3\le r,k\le n$. The bounds differ for different ranges of r compared to n and k.

A graph is ℓ-holed if all its induced cycles of length at least four have length exactly ℓ. We give a complete description of the ℓ-holed graphs for each $\ell \ge 7$.

We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are $P_2$ and $P_3$. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.