Related papers: Clumsy packings of graphs

Given two graphs $G$ and $H$, a $(G,H)$-multidecomposition of $K_{n}$ is a partition of the edges of $K_{n}$ into copies of $G$ and $H$ such that at least one copy of each is used. We give necessary and sufficient conditions for the…

We consider the following generalization of dominating sets: Let $G$ be a host graph and $P$ be a pattern graph $P$. A dominating $P$-pattern in $G$ is a subset $S$ of vertices in $G$ that (1) forms a dominating set in $G$ \emph{and} (2)…

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$,…

A drawing of a graph in the plane is called 1-planar if each edge is crossed at most once. A graph together with a 1-planar drawing is a 1-plane graph. A 1-plane graph $G$ with exactly $4|V (G)|-8$ edges is called optimal. The crossing…

For a graph $H$, let $$c_{\infty}(H)= \lim_{n \to \infty}\max\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\infty}(H)$ is the asymptotic maximum density of graphs…

The packing chromatic number $\chi_\rho(G)$ of a graph $G$ is the smallest integer $k$ needed to proper color the vertices of $G$ in such a way the distance between any two vertices having color $i$ be at least $i+1$. We obtain…

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs $H$ of a graph $G$ of the neighborhood set system of $H$ is sandwiched…

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest…

Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(\chi(H) - 1) + \sigma(H)$, where $\chi(H)$ denotes the usual chromatic number of $H$, and $\sigma(H)$ denotes…

Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey…

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of…

The maximum genus $\gamma_M(G)$ of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal…

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains…

For a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, a packing $S$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, V_2, \ldots, V_k$ such that each $V_i$ has pairwise distance at least $s_i+1$. The…

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and…

A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine…

The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In…

For a non-decreasing sequence of integers $S=(s_1,s_2, \dots, s_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at…

An old conjecture of Zs. Tuza says that for any graph $G$, the ratio of the minimum size, $\tau_3(G)$, of a set of edges meeting all triangles to the maximum size, $\nu_3(G)$, of an edge-disjoint triangle packing is at most 2. Here,…

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c…