European Journal of Combinatorics最新文献

We prove that every locally finite quasi-transitive graph that does not contain $K_{\infty }$ as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their recent result that such graphs are quasi-isometric to some planar graph of bounded degree.

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$, and an $H$-structure $h$ on a set $N$, there are proper colorings of $h$, generalizing graph colorings and poset partitions. We show that the automorphism group of $h$ acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.

We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs $G$ such that any 2-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of $G$ (that is, a copy with half the edges of each color). While the problem of recognizing balanceable graphs was conjectured to be $\mathrm{NP}$-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph’s edges and another one such that the corresponding cut contains half of all the graph’s edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks’ theorem, we show that every 4-regular graph with order divisible by 4 is balanceable. Finally, we show that it is $\mathrm{NP}$-complete to determine if a 9-regular graph is simply balanceable.

The pop-stack-sorting process is a variation of the stack-sorting process. We consider a deterministic version of this process. We prove a lemma which characterises interior elements of increasing runs after $t$ iterations of the process and provide a new lower bound of $\frac{3}{5}n$ for the number of iterations of the process to fully sort a uniformly randomly chosen permutation of length $n$.

Nash-Williams proved in 1960 that a finite graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected, and conjectured that the same result should hold for all infinite graphs, too.

Progress on Nash-Williams’s problem was made by C. Thomassen, who proved in 2016 that all $8k$-edge-connected infinite graphs admit a $k$-arc connected orientation, and by the first author, who recently showed that edge-connectivity of $4k$ suffices for locally-finite, 1-ended graphs.

In the present article, we establish the optimal bound $2k$ in Nash-Williams’s conjecture for all locally finite graphs with countably many ends.

The analogue of Hadwiger’s conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{\chi \left(G\right)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n/2\rceil }$. We show that every $n$-vertex graph with independence number 2 contains every complete bipartite graph on $\lceil n/2\rceil$ vertices as an immersion.

Let $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ be an intersecting family of sets and let $\Delta \left(F\right)$ be the maximum degree in $F$, i.e., the maximum number of edges of $F$ containing a fixed vertex. The diversity of $F$ is defined as $d\left(F\right)≔\left|F\right|-\Delta \left(F\right)$. Diversity can be viewed as a measure of distance from the ‘trivial’ maximum-size intersecting family given by the Erdős–Ko–Rado Theorem. Indeed, the diversity of this family is 0. Moreover, the diversity of the largest non-trivial intersecting family, due to Hilton–Milner, is 1. It is known that the maximum possible diversity of an intersecting family $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ is $\left(\frac{n-3}{r-2}\right)$ as long as $n$ is large enough.

We introduce a generalization called the $C$-weighted diversity of $F$ as $d_C\left(F\right)≔\left|F\right|-C\cdot \Delta \left(F\right)$. We determine the maximum value of $d_C\left(F\right)$ for intersecting families $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ and characterize the maximal families for $C\in \left(0,\frac{7}{3}\right)$ as well as give general bounds for all $C$. Our results imply, for large $n$, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl’s Delta-system method.

The domination game is an optimization game played by two players, Dominator and Staller, who alternately select vertices in a graph $G$. A vertex is said to be dominated if it has been selected or is adjacent to a selected vertex. Each selected vertex must strictly increase the number of dominated vertices at the time of its selection, and the game ends once every vertex in $G$ is dominated. Dominator aims to keep the game as short as possible, while Staller tries to achieve the opposite. In this article, we prove that for any graph $G$ on $n$ vertices, Dominator has a strategy to end the game in at most $3n/5$ moves, which was conjectured by Kinnersley, West and Zamani.

We prove that for every $k\ge 10$, the online Ramsey number for paths $P_k$ and $P_n$ satisfies $\tilde{r}(P_k,P_n)\ge \frac{5}{3}n+\frac{k}{9}-4$, matching up to a linear term in $k$ the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies ${lim}_{n\to \infty }\frac{\tilde{r}(P_k,P_n)}{n}=\frac{5}{3}$, whenever $10\le k=o\left(n\right)$, disproving a conjecture by Cyman et al. (2015).

A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $R$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and results about direct sums of positroids involve connected flats and non-crossing partitions. We prove another characterization of positroids of a similar flavor and give some applications of the characterization. We show that if $M$ and $N$ are positroids and the intersection of their ground sets is an independent set and a set of clones in both $M$ and $N$, then the free amalgam of $M$ and $N$ is a positroid, and we prove a second result of that type. Also, we identify several multi-parameter infinite families of excluded minors for the class of positroids.