Related papers: Low Diameter Monochromatic Covers of Complete Mult…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$…

The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define…

Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$.…

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool…

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…

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a…

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to…

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the…

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of…

For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the…

A well-known conjecture by Harris states that any triangle-free $d$-degenerate graph has fractional chromatic number at most $O\left(\frac{d}{\ln d}\right)$. This conjecture has gained much attention in recent years, and is known to have…

The biclique cover number (resp. biclique partition number) of a graph $G$, $\mathrm{bc}(G$) (resp. $\mathrm{bp}(G)$), is the least number of biclique (complete bipartite) subgraphs that are needed to cover (resp. partition) the edges of…

Given a connected and bridgeless graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$. The orientation number of $G$ is defined to be $\bar{d}(G):=min\{d(D)|D\in \mathscr{D}(G)\}$, where $d(D)$ is the diameter of the…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

Let $G$ be a graph with minimum degree at least 2. A set $D\subseteq V$ is a double total dominating set of $G$ if each vertex is adjacent to at least two vertices in $D$. The double total domination number $\gamma _{\times 2,t}(G)$ of $G$…

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a…

The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c))…

Inspired by the phenomenon of co-channel interference in communication network, a novel graph parameter, called $r$-vertex-strongly-distinguishing total coloring (abbreviate as $D(r)$-VSDTC), is proposed in this paper. Given a graph $G$, an…