European Journal of Combinatorics最新文献

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.

In this note we show that the singular probability of the adjacency matrix of a random $d$-regular graph on $n$ vertices, where $d$ is fixed and $n\to \infty$, is bounded by $n^{-1/3+o\left(1\right)}$. This improves a recent bound by Huang in Huang (2021). Our method is based on the study of the singularity problem modulo a prime developed in Huang (2021) (and also partially in Mészáros, 2021; Nguyen and Wood, 2018), together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser’s problem in combinatorics.

An edge $e$ is normal in a proper edge-coloring of a cubic graph $G$ if the number of distinct colors on four edges incident to $e$ is 2 or $4.$ A normal edge-coloring of $G$ is a proper edge-coloring in which every edge of $G$ is normal. The Petersen Coloring Conjecture is equivalent to stating that every bridgeless cubic graph has a normal 5-edge-coloring. Since every 3-edge-coloring of a cubic graph is trivially normal, it is sufficient to consider only snarks to establish the conjecture. In this paper, we consider a class of superpositioned snarks obtained by choosing a cycle $C$ in a snark $G$ and superpositioning vertices of $C$ by one of two simple supervertices and edges of $C$ by superedges $H_{x,y}$, where $H$ is any snark and $x,y$ any pair of nonadjacent vertices of $H.$ For such superpositioned snarks, two sufficient conditions are given for the existence of a normal 5 -edge-coloring. The first condition yields a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but only for some of the possible ways of connecting them. In particular, since the Flower snarks are hypohamiltonian, this consequently yields a normal 5-edge-coloring for many snarks superpositioned by the Flower snarks. The second sufficient condition is more demanding, but its application yields a normal 5-edge-colorings for all superpositions by the Flower snarks. The same class of snarks is considered in Liu et al. (2021) for the Berge–Fulkerson conjecture. Since we established that this class has a Petersen coloring, this immediately yields the result of the above mentioned paper.

The Odd Hadwiger’s conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger’s famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic number, introduced by Matoušek and Ziegler (2003) and refined recently by Daneshpajouh and Meunier (2023), forms a potential avenue to a disproof of Hadwiger’s conjecture or its odd-minor variant. In this direction, we prove that, in a very general sense, every graph $G$ that admits a topological lower bound of $t$ on its chromatic number, contains $K_{\lfloor t/2\rfloor +1}$ as an odd-minor. This solves a problem posed by Simonyi and Zsbán (2010).

We also prove that if for a graph $G$ the Dol’nikov-Kříž lower bound on the chromatic number (one of the lower bounds in the aforementioned hierarchy) attains a value of at least $t$, then $G$ contains $K_t$ as a minor.

Finally, extending results by Simonyi and Zsbán, we show that the Odd Hadwiger’s conjecture holds for Schrijver and Kneser graphs for any choice of the parameters. The latter are canonical examples of graphs for which topological lower bounds on the chromatic number are tight.

Let $G$ be a graph with maximum degree $\Delta \left(G\right)$ and maximum multiplicity $\mu \left(G\right)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta \left(G\right)+\mu \left(G\right)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Albertson and Moore conjectured that if $G$ is a simple graph, using the palette $\{1,\dots ,\Delta \left(G\right)+1\}$, any precoloring on a distance-3 matching can be extended to a proper edge coloring of $G$. Edwards et al. proposed the following stronger conjecture: For any graph $G$, using the palette $\{1,\dots ,\Delta \left(G\right)+\mu \left(G\right)\}$, any precoloring on a distance-2 matching can be extended to a proper edge coloring of $G$. Girão and Kang verified the conjecture of Edwards et al. for distance-9 matchings. In this paper, we improve the required distance from 9 to 3 for multigraphs $G$ with $\mu \left(G\right)\ge 2$.

Sivaraman (2020) conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1,X_2\subseteq V\left(G\right)$ satisfying $V\left(G\right)=X_1\cup X_2$ such that the induced graphs $G\left[X_1\right]$ and $G\left[X_2\right]$ are both chordal. We prove this conjecture in the special case where $G$ contains no sector wheel, namely, a pair $(H,w)$ where $H$ is an induced cycle of $G$ and $w$ is a vertex in $V\left(G\right)\setminus V\left(H\right)$ such that $N\left(w\right)\cap H$ is either $V\left(H\right)$ or a path with at least three vertices.