If S = (a 1 , a 2 , . . .) is a non-decreasing sequence of positive integers, then an S-packing coloring of a graph G is a partition of V (G) into sets X 1 , X 2 , . . . such that for each pair of distinct vertices in the set X i , the distance between them is larger than a i . If there exists an integer k such that

A classical result of Erdős and Hajnal claims that for any integers k, r, g ≥ 2 there is an r-uniform hypergraph of girth at least g with chromatic number at least k. This implies that there are sparse hypergraphs such that in any coloring of their vertices with at most k − 1 colors there is a monochromatic hyperedge. When there is no restriction on the number of the colors used, one can easily avoid monochromatic hyperedges. Then, however, so-called rainbow or multicolored hyperedges might appear. Nešetřil and Rödl [19] called hypergraphs such that in any vertex-coloring there is either a monochromatic or a rainbow hyperedge, selective. They showed an existence of selective r-uniform hypergraphs of girth g for any integers r, g ≥ 2 using probabilistic and explicit constructions. In this paper, we provide a slightly di erent construction of such hypergraphs and summarize the probabilistic approaches. The main building block of the construction, a part-rainbow-forced hypergraph, is of independent interest. This is an r-uniform r-partite hypergraph with a given girth such that in any vertex-coloring that is rainbow on each part, there is a rainbow hyperedge. We give a simple construction of such a hypergraph that does not use iterative amalgamation.

Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies that vertices $u$ and $v$ are at distance greater than $i$, is called a packing coloring of $G$. The minimum number of colors in a packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $\chi_\rho(G)$. In this paper, we propose the study of $\chi_\rho$-critical graphs, which are the graphs $G$ such that for any proper subgraph $H$ of $G$, $\chi_\rho(H)<\chi_\rho(G)$. We characterize $\chi_\rho$-critical graphs with diameter 2, and $\chi_\rho$-critical block graphs with diameter 3. Furthermore, we characterize $\chi_\rho$-critical graphs with small packing chromatic numbers, and we also consider $\chi_\rho$-critical trees. In addition, we prove that for any graph $G$ with $e\in E(G)$, we have $(\chi_\rho(G)+1)/2\le \chi_\rho(G-e)\le \chi_\rho(G)$, and provide a corresponding realization result, which shows that $\chi_\rho(G-e)$ can achieve any of the integers between the bounds.

Let D = (V,A) be a digraph of order n, S a subset of V of size k and 2 ≤ k ≤ n. A subdigraph H of D is called an S-strong subgraph if H is strong and S ⊆ V (H). Two S-strong subgraphs D1 and D2 are said to be arc-disjoint if A(D1) ∩ A(D2) = ∅. Let λS(D) be the maximum number of arc-disjoint S-strong digraphs in D. The strong subgraph k- arc-connectivity is defined as λk(D) = min{λS(D) | S ⊆ V, |S| = k}. A digraph D = (V,A) is called minimally strong subgraph (k, ℓ)-arc-connected if λk(D) ≥ ℓ but for any arc e ∈ A, λk(D − e) ≤ ℓ − 1. Let 𝔊 (n, k, ℓ) be the set of all minimally strong subgraph (k, ℓ)-arc-connected digraphs with order n. We define G(n, k, ℓ) = max{|A(D)| | D ∈𝔊 (n, k, ℓ)} and g(n, k, ℓ) = min{|A(D)| | D ∈ 𝔊(n, k, ℓ)}.

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted sub k (G). This invariant serves as a generalization of the Slater number ; in particular, we show that sub k (G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination numbers of several families of graphs, provide structural results on sub-kdomination, and explore properties of graphs which are sub k (G)-critical with respect to addition and deletion of vertices and edges.

Under an edge coloring (not necessarily proper), a rainbow path is a path whose edge colors are all distinct. The d-local rainbow connection number lrcd(G) (respectively, d-local strong rainbow connection number lsrcd(G)) is the smallest number of colors needed to color the edges of G such that any two vertices with distance at most d can be connected by a rainbow path (respectively, rainbow geodesic). This generalizes rainbow connection numbers, which are the special case d = diam(G). We discuss some bounds and exact values. Moreover, we also characterize all triples of positive integers d, a, b such that there is a connected graph G with lrcd(G) = a and lsrcd(G) = b.

A graph is called Hamiltonian extendable if there exists a Hamiltonian path between any two nonadjacent vertices. In this paper, we give an explicit formula of the minimum number of edges for Hamiltonian extendable graphs and we also characterize the degree sequence for Hamiltonian extendable graphs with minimum number of edges. Besides, we completely characterize the pairs of forbidden subgraphs for 2-connected graphs to be Hamiltonian extendable.

A bipartite-split graph is a bipartite graph whose vertex set can be partitioned into a complete bipartite set and an independent set. The bipartitesplittance of an arbitrary bipartite graph is the minimum number of edges to be added or removed in order to produce a bipartite-split graph. In this paper, we show that the bipartite-splittance of a bipartite graph depends only on the degree sequence pair of the bipartite graph, and an easily computable formula for it is derived. As a corollary, a simple characterization of the degree sequence pair of bipartite-split graphs is also given.

Let G be a cellular embedding of a multigraph in a 2-manifold. Two distinct edges e 1 , e 2 E(G) are facially adjacent if they are consecutive on a facial walk of a face f F (G). An incidence of the multigraph G is a pair (v, e), where v V (G), e E(G) and v is incident with e in G. Two distinct incidences (v 1 , e 1 ) and (v 2 , e 2 ) of G are facially adjacent if either e 1 = e 2 or e 1 , e 2 are facially adjacent and either v 1 = v 2 or v 1 = v 2 and there is i {1, 2} such that e i is incident with both v 1 , v 2 . A facial incidence coloring of G assigns a color to each incidence of G in such a way that facially adjacent incidences get distinct colors. In this note we show that any embedded multigraph has a facial incidence coloring with seven colors. This bound is improved to six for several wide families of plane graphs and to four for plane triangulations.

A (graph) property P is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties P.