相关论文: On a Generalization of Tupper's Formula for $m$ Co…

The purpose of this note is to draw attention to problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised a problem of determining, for a natural number $k$, the…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we…

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi…

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many…

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\leq n.$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+{k+1 \choose 2}.$ In this paper we…

A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for…

An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an…

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known…

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total…

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This…

A $1$-factorization of the complete multigraph $\lambda K_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of $\lambda_0 K_{2n}$ and $(\lambda-\lambda_0) K_{2n}$, where $\lambda_0<\lambda$.…

We consider the problem of existence of perfect $2$-colorings in the Doob graphs $D(m,n)$ and $4$-ary Hamming graphs $H(n,4)$. We characterize all parameters for which multifold $1$-perfect code in $D(m,n)$ exists. Also, we prove that for…

The concept of metric dimension has applications in a variety of fields, such as chemistry, robotic navigation, and combinatorial optimization. We show bounds for graphs with $n$ vertices and metric dimension $\beta$. For Hamiltonian…

Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson…

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The…

We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for…

It is proved that for each natural number $n$, if $\left| \mathbb{R} \right| = {\aleph}_{n}$, then there is a coloring of ${\left[ \mathbb{R} \right]}^{n+2}$ into ${\aleph}_{0}$ colors that takes all colors on ${\left[ X \right]}^{n+2}$…