A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph $K_{k+1}$ by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós’ theorem to list-colorings of hypergraphs.

By a hamiltonian coloring of a connected graph $G$ of order $n\ge 1$ we mean a mapping $c$ of $V\left(G\right)$ into the set of all positive integers such that $|c\left(x\right)-c\left(y\right)|\ge n-1-D_G(x,y)$ (where $D_G(x,y)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x,y\in G$. In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of order $n\ge 5$ with circumference $n-2$.

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^k$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^k$ if the directed distance ${}^{\to }{d}_D(u,v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^k$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^k$ is distance-colored if each arc (u, v) of $D^k$ is assigned the color i where $i{=}^{\to }{d}_D(u,v)$. The digraph $D^k$ is Hamiltonian-colored...

The total generalised colourings considered in this paper are colourings of graphs such that the vertices and edges of the graph which receive the same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total chromatic number is the least number of colours with which this is possible. We study such colourings for sets of planar graphs and determine, in particular, upper bounds for these chromatic numbers for proper...

A total-colored path is total rainbow if both its edges and internal vertices have distinct colors. The total rainbow connection number of a connected graph G, denoted by trc(G), is the smallest number of colors that are needed in a total-coloring of G in order to make G total rainbow connected, that is, any two vertices of G are connected by a total rainbow path. In this paper, we study the computational complexity of total rainbow connection of graphs. We show that deciding whether a given total-coloring...

By means of branched coverings techniques, we prove that the Heegaard genus and the regular genus of an orientable 3-manifold with boundary coincide.

A 2-stratified graph $G$ is a graph whose vertex set has been partitioned into two subsets, called the strata or color classes of $G$. Two $2$-stratified graphs $G$ and $H$ are isomorphic if there exists a color-preserving isomorphism $\phi$ from $G$ to $H$. A $2$-stratified graph $G$ is said to be homogeneously embedded in a $2$-stratified graph $H$ if for every vertex $x$ of $G$ and every vertex $y$ of $H$, where $x$ and $y$ are colored the same, there exists an induced $2$-stratified subgraph $H^{\text{'}}$ of $H$ containing $y$ and a color-preserving...

In this note it is shown that every hypergraph containing a pendant path of length at least 2 is not chromatically unique. The same conclusion holds for h-uniform r-quasi linear 3-cycle if r ≥ 2.