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.

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.

We prove: (1) that $c{h}_P\left(G\right)-{\chi }_P\left(G\right)$ can be arbitrarily large, where $c{h}_P\left(G\right)$ and ${\chi }_P\left(G\right)$ are P-choice and P-chromatic numbers, respectively, (2) the (P,L)-colouring version of Brooks’ and Gallai’s theorems.

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number ${\chi }_D\left(G\right)$ of G. Extending these concepts to infinite graphs we prove that $D\left(Q_{\aleph }₀\right)=2$ and ${\chi }_D\left(Q_{\aleph }₀\right)=3$, where $Q_{\aleph }₀$ denotes the hypercube of countable dimension. We also show that ${\chi }_D\left(Q₄\right)=4$, thereby completing the investigation of finite hypercubes with respect to ${\chi }_D$. Our...

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...

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$, a hereditary graph property $𝒫$ and $l\ge 1$ we define ${\chi }_{𝒫,l}^{\text{'}}\left(G\right)$ to be the minimum number of colours needed to properly colour the edges of $G$, such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $𝒫$. We present a necessary and sufficient condition for the existence of ${\chi }_{𝒫,l}^{\text{'}}\left(G\right)$. We focus on edge-colourings of graphs with respect to the hereditary...

A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph ${}_{d̲,\left[d̅\right]}\left(S\right)=(V,)$ where V = S and $=e\subseteq S:d̲\le \left|e\right|\le \left[d̅\right]\wedge {\sum }_{v\in e}v\in S$. For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $y₁,...,y_{\sigma }\notin V$ such that $\cup y₁,...,y_{\sigma }$ is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings...

In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g(n,l)$, the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$. By edit distance of two graphs $G$, $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$. This new extremal number $g(n,l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition $E_{All}$, $E_{Some}$, $E_{None}$ of the edge set E, where: $E_{All}$ = e ∈ E, e belongs to all optimum solutions, $E_{None}$ = e ∈ E, e does not belong to any optimum solution and $E_{Some}$ = e ∈ E, e belongs to some but not to all optimum solutions.

The generalized $k$-connectivity ${\kappa }_k\left(G\right)$ of a graph $G$ was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as ${\lambda }_k\left(G\right)=min\{\lambda \left(S\right):S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$, where $\lambda \left(S\right)$ denotes the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,...,T_{\ell }$ in $G$ such that $S\subseteq V\left(T_i\right)$ for $1\le i\le \ell$. In this paper we prove that for any two connected graphs $G$ and $H$ we have ${\lambda }_3(G\square H)\ge {\lambda }_3\left(G\right)+{\lambda }_3\left(H\right)$, where $G\square H$ is the Cartesian product of $G$ and $H$. Moreover, the bound is sharp. We also...

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left(S_{p,q}\right)$ and $va{l}_{min}\left(S_{p,q}\right)$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which...

For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁\times ...\times G_N$ is embeddable into ${ℝ}^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $\left(K_{3,3}\right)ⁿ$ are not embeddable into ${ℝ}^{2n}$, where K₅ and $K_{3,3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $LkO\to S^{2n-1}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

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....

We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $\left|lk\right(Q_i,Q_j\left)\right|\ge \alpha$ and $\left|a₂\right(Q_i\left)\right|\ge \alpha$, where $a₂\left(Q_i\right)$ denotes the second coefficient of the Conway polynomial of $Q_i$.

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

A vertex $k$-coloring of a graph $G$ is a if $M\left(u\right)\ne M\left(v\right)$ for every edge $uv\in E\left(G\right)$, where $M\left(u\right)$ and $M\left(v\right)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the ${\chi }_m\left(G\right)$ of $G$. For an integer $\ell \ge 0$, the $\ell$- of a graph $G$, ${\mathrm{cor}}^{\ell }\left(G\right)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for $\ell$- of all complete graphs, the regular complete...

A graph is called distance integral (or $D$-integral) if all eigenvalues of its distance matrix are integers. In their study of $D$-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_1,p_2,p_3}$ with $p_1<p_2<p_3$, and $K_{p_1,p_2,p_3,p_4}$ with $p_1<p_2<p_3<p_4$, as well as the infinite classes of distance integral...

A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each...