Related papers: Note on parity factors of regular graphs

An equation is called graph-regular if it always has monochromatic solutions under edge-colorings of the complete graph on the naturals. We present two Rado-like conditions which are respectively necessary and sufficient for an equation to…

In this paper, we discuss the existence of fixed points for integral type contractions in uniform spaces endowed with both a graph and an $E$-distance. We also give two sufficient conditions under which the fixed point is unique. Our main…

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and…

We introduce the concept of matching connectivity as a notion of connectivity in graph admitting perfect matchings which heavily relies on the structural properties of those matchings. We generalise a result of Robertson, Seymour and Thomas…

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or…

In this paper we study a new variant of graph arboricity, which requires all the forests to have the same number of edges (up to a difference of 1). We prove that the new variant, which we call equitable arboricity, is equivalent to…

Haxell's condition is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the…

Graph pooling is commonly applied in graph classification, yet its empirical gains over standard WL-1 expressive GNNs are often marginal or inconsistent. We study this gap by analysing the interaction between node features and graph…

We derive a new upper bound on the algebraic connectivity of a regular graph using the Higman-Sims technique. Together with a new result on the connectivity of the neighbourhood graph of strongly regular graphs, our result gives a…

A graph $G$ is embeddable in $\mathbb{R}^d$ if vertices of $G$ can be assigned with points of $\mathbb{R}^d$ in such a way that all pairs of adjacent vertices are at the distance 1. We show that verifying embeddability of a given graph in…

Addressing a question posed by Chen and Ma from an asymptotic point of view, we present a short proof for the edge density needed to guarantee that two vertices of the same degree are connected by a path of a fixed length. In particular, we…

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of…

We derive explicit formulas for the Lin-Lu-Yau curvature and the Ollivier-Ricci curvature in terms of graph parameters and an optimal assignment. Utilizing these precise expressions, we examine the relationship between the Lin-Lu-Yau…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we…

Let R be an equivalence relation on graphs. By the strengthening of R we mean the relation R' such that graphs G and H are in the relation R' if for every graph F, the union of the graphs G and F is in the relation R with the union of the…

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow \{0,1,\ldots, k\}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ for every vertex $v \in V(G)$,…

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide, to the best of our knowledge, the first characterization of…

Given an edge-coloring of a graph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. We define the palette index of a graph as the minimum number of distinct palettes, taken over all…

We consider quantum search on graphs. Recently, it has been shown that the graph properties like connectivity, global symmetry, or regularity cannot serve as a reliable criteria that must be satisfied by a graph to allow a successful…

This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these…