Related papers: On Alternating 6-Cycles in Edge-Coloured Graphs
Let $H$ be a $k$-edge-coloured graph and let $n$ be a positive integer. What is the maximum number of copies of $H$ in a $k$-edge-coloured complete graph on $n$ vertices? This paper studies the case $k=2$, which we call the…
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph…
We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are…
Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let $HP^{(\k)}_{n,m,k}$ be an edge colored version, where we color each edge randomly from…
We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…
A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs,…
We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…
We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint…
Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden…
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…
We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $H$ is a $k$-uniform hypergraph with minimum codegree at least $(1/2 + \gamma )n$, $\gamma >0$, and $n$ is sufficiently large,…
We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so that the blue and red graphs are connected and the blue and red degrees at each vertex differ by at most four. This improves a…
We propose an effective algorithm that enumerates (and actually finds) all 3-edge colorings and Hamiltonian cycles in a cubic graph. The idea is to make a preliminary run that separates the vertices into two types: ``rigid'' (such that the…
In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out…
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…
A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…
A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show…
For a graph $H$ whose edges are coloured blue or red, the $H$-semi-inducibility problem asks for the maximum, over all graphs $G$ of given order $n$, of the number of injections from the vertex set of $H$ into the vertex set of $G$ that…
In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our…
We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G))…