Related papers: Homogeneous coloured multipartite graphs

A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums…

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a…

We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…

An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph homomorphism of an $(m, n$)-colored mixed graph $G$ to an $(m, n)$-colored mixed graph $H$ is a vertex mapping such…

Unigraphs are graphs identifiable up to isomorphism from their degree sequences. Given a class $\mathcal{A}$ of graphs, we define the class of $\mathcal{A}$-unigraphs to be graphs identifiable from degree sequence and membership in…

For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as…

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two…

We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and…

A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of…

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

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…

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e.,…

It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of…

The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.

We introduce and study Fair and Tolerant colorings (FAT colorings), where each vertex tolerates a given fraction of same-colored neighbors while fairness is preserved across the other coloring classes. Moreover, we define the FAT chromatic…

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We…

This is the first in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…