A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph ${⁺}_d\left(S\right)=(V,)$, where V = S and $=v₁,...,v_d:\left(i\ne j\Rightarrow v_i\ne v_j\right)\wedge {\sum }_{i=1}^d{v}_i\in S$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w₁,...,w_{\sigma }\notin V$ such that $\cup w₁,...,w_{\sigma }$ is a sum hypergraph. In this paper, we prove $\sigma {(}_{n₁,...,n_d}^d)=1+{\sum }_{i=1}^d(n_i-1)+min0,\lceil 1/2({\sum }_{i=1}^{d-1}(n_i-1)-n_d)\rceil$, where ${}_{n₁,...,n_d}^d$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

A hypergraph ℋ is a sum hypergraph iff there are a finite S ⊆ ℕ⁺ and d̲,d̅ ∈ ℕ⁺ with 1 < d̲ < d̅ such that ℋ is isomorphic to the hypergraph ${\mathscr{H}}_{d̲,d̅}\left(S\right)=(V,)$ where V = S and $=e\subseteq S:d̲<\left|e\right|<d̅\wedge {\sum }_{v\in e}v\in S$. For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w₁,...,w_{\sigma }\notin V$ such that $\mathscr{H}\cup w₁,...,w_{\sigma }$ is a sum hypergraph. For graphs it is known that cycles Cₙ and wheels Wₙ have sum numbersgreater than one. Generalizing these graphs we prove for the hypergraphs ₙ and ₙ that under a certain condition...

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers $s_i$ and $t_i$ satisfying $1\le s_i\le t_i\le \left|E_i\right|$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_i$ satisfies $s_i\le \left|\phi \left(E_i\right)\right|\le t_i$. The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each $E_i$ induces a connected subgraph in G....

Let $G$ be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that $G$ admits a bipartition such that each vertex class meets edges of total weight at least $(w_1-{\Delta }_1)/2+2w_2/3$, where $w_i$ is the total weight of edges of size $i$ and ${\Delta }_1$ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph $G$ (i.e., multi-hypergraph), we show that there exists a bipartition of $G$ such that each vertex class meets edges of total weight at least $(w_0-1)/6+(w_1-{\Delta }_1)/3+2w_2/3$, where...

We consider random dynamics on the edges of a uniform Cayley tree with $n$ vertices, in which edges are either flammable, fireproof, or burnt. Every flammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate $n^{-\alpha }$ on each flammable edge, then propagate through the neighboring flammable edges and are only stopped at fireproof edges. A vertex is called fireproof when all its adjacent edges are fireproof. We show that as $n\to \infty$, the terminal density of fireproof...

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple graph and $E_G\left(v\right)$ denote the set of edges incident with a vertex $v$. A neighbor sum distinguishing (NSD) total coloring $\phi$ of $G$ is a proper total coloring of $G$ such that ${\sum }_{z\in E_G\left(u\right)\cup \left\{u\right\}}\phi \left(z\right)\ne {\sum }_{z\in E_G\left(v\right)\cup \left\{v\right\}}\phi \left(z\right)$ for each edge $uv\in E\left(G\right)$. Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $\Delta$ admits an NSD total $(\Delta +3)$-coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $\Delta \ge 11$ but without $5$-cycles by applying the Combinatorial Nullstellensatz.

Let $G$ be a simple graph, let $d\left(v\right)$ denote the degree of a vertex $v$ and let $g$ be a nonnegative integer function on $V\left(G\right)$ with $0\le g\left(v\right)\le d\left(v\right)$ for each vertex $v\in V\left(G\right)$. A $g_c$-coloring of $G$ is an edge coloring such that for each vertex $v\in V\left(G\right)$ and each color $c$, there are at least $g\left(v\right)$ edges colored $c$ incident with $v$. The $g_c$-chromatic index of $G$, denoted by ${\chi }_{g_c}^{\text{'}}\left(G\right)$, is the maximum number of colors such that a $g_c$-coloring of $G$ exists. Any simple graph $G$ has the $g_c$-chromatic index equal to ${\delta }_g\left(G\right)$ or ${\delta }_g\left(G\right)-1$, where ${\delta }_g\left(G\right)={min}_{v\in V\left(G\right)}\lfloor d\left(v\right)/g\left(v\right)\rfloor$. A graph $G$ is nearly bipartite,...

We consider, for a positive integer $k$, induced subgraphs in which each component has order at most $k$. Such a subgraph is said to be $k$-divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a $k$-divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for...

For a digraph $D$, the niche hypergraph $N\mathscr{H}\left(D\right)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges $E\left(N\mathscr{H}\right(D\left)\right)=\{e\subseteq V(D):|e|\ge 2$ and there exists a vertex $v$ such that $e=N_D^{-}\left(v\right)$ or $e=N_D^{+}\left(v\right)\}$. A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $\mathscr{H}$, the niche number $\widehat{n}\left(\mathscr{H}\right)$ is the smallest integer such that $\mathscr{H}$ together with $\widehat{n}\left(\mathscr{H}\right)$ isolated vertices is the niche hypergraph of an acyclic digraph. C. Garske, M. Sonntag and H. M. Teichert (2016) conjectured that for a linear...

A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle in $G$. In other words, each cycle of $G$ must be colored with at least three colors. Given a list assignment $L=\left\{L\right(v):v\in V\}$, if there exists an acyclic coloring $\pi$ of $G$ such that $\pi \left(v\right)\in L\left(v\right)$ for all $v\in V$, then we say that $G$ is acyclically $L$-colorable. If $G$ is acyclically $L$-colorable for any list assignment $L$ with $\left|L\right(v\left)\right|\ge k$ for all $v\in V$, then $G$ is acyclically $k$-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without...

A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number $\chi {*}_r\left(G\right)$ of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that $\chi {*}_r\left(Pₙ\right)<3log₂n$ for n ≥ 2....

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$...

An edge of $G$ is singular if it does not lie on any triangle of $G$; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G\left[N\right(u\left)\right]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such...

The Turán number of a given graph $H$, denoted by $\mathrm{ex}(n,H)$, is the maximum number of edges in an $H$-free graph on $n$ vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number $\text{ex}(n,K_p\cup K_q$) of a vertex-disjoint union of cliques $K_p$ and $K_q$ for all values of $n$.