Related papers: Monochromatic Clique Decompositions of Graphs

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$…

For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…

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…

A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present…

We study the problem of partitioning the edges of a $d$-uniform hypergraph $H$ into a family $F$ of complete $d$-partite hypergraphs ($d$-cliques). We show that there is a partition $F$ in which every vertex $v \in V(H)$ belongs to at most…

For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a…

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We…

Fix an integer $k \ge 3$. A $k$-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant $c$ depending only on $k$ such that every simple $k$-uniform hypergraph $H$ with maximum degree $\D$…

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…

Fix $r \ge 2$ and a collection of $r$-uniform hypergraphs $\cH$. What is the minimum number of edges in an $\cH$-free $r$-uniform hypergraph with chromatic number greater than $k$. We investigate this question for various $\cH$. Our results…

Given a graph $G$, the $k$-coloring graph $\mathcal{C}_k(G)$ is constructed by selecting proper $k$-colorings of $G$ as vertices, with an edge between two colorings if they differ in the color of exactly one vertex. The number of vertices…

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any…

The $H$-Coloring problem is a well-known generalization of the classical NP-complete problem $k$-Coloring where the task is to determine whether an input graph admits a homomorphism to the template graph $H$. This problem has been the…

Given an $r$-edge-coloring of the complete graph $K_n$, what is the largest number of edges in a monochromatic connected component? This natural question has only recently received the attention it deserves, with work by two disjoint…

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be…

The harmonious chromatic number of a graph $G$ is the minimum number of colors that can be assigned to the vertices of $G$ in a proper way such that any two distinct edges have different color pairs. This paper gives various results on…

We prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $\frac m{n-1}$ is an integer and the maximum degree of $K$ is at most $\frac {2m}{n-1}$.

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all…