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...

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...

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.

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2,2,r}$ r ∈ 4,5,6,7,8, $K_{2,3,4}$, $K_{1*4,4}$, $K_{1*4,5}$, $K_{1*5,4}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2,2,r}$ r ∈ 4,5,6,7,8, the others have been proved not...

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...

In this paper, the chromaticity of K₃-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K₃-gluings of wheels $W_{q+2}$ and $W_{n-q+2}$ create an χ-equivalent class.

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 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...

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 $Q_i$.

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 graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba’s conjecture is true for complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $ch\left(K_{4,3*t,2*(k-2t-2),1*(t+1)}\right)=k$, which extends the results $ch\left(K_{4,3,2*(k-4),1*2}\right)=k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $ch\left(K_{4,3*2,2*(k-6),1*3}\right)=k$...

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...

We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $\left(\frac{N}{2}\right)-o\left(N^2\right)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o\left(1\right)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-\delta }$ induced matchings, where $\delta >0,076$. This disproves (in a strong form) a conjecture of Meshulam,...

The paired domination number ${\gamma }_{pr}\left(G\right)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: ${\gamma }_{pr}\left(\pi G\right)=2{\gamma }_{pr}\left(G\right)$ for all πG; ${\gamma }_{pr}\left(K₂☐G\right)=2{\gamma }_{pr}\left(G\right)$; ${\gamma }_{pr}\left(K₂☐G\right)={\gamma }_{pr}\left(G\right)$.

Let $\gamma \left(G\right)$ and $i\left(G\right)$ be the domination number and the independent domination number of $G$, respectively. Rad and Volkmann posted a conjecture that $i\left(G\right)/\gamma \left(G\right)\le \Delta \left(G\right)/2$ for any graph $G$, where $\Delta \left(G\right)$ is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $\Delta \left(G\right)/2$ are provided as well.

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.

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.

For a set D of positive integers, we define a vertex set S ⊆ V(G) to be D-independent if u, v ∈ S implies the distance d(u,v) ∉ D. The D-independence number ${\beta }_D\left(G\right)$ is the maximum cardinality of a D-independent set. In particular, the independence number $\beta \left(G\right)={\beta }_1\left(G\right)$. Along with general results we consider, in particular, the odd-independence number ${\beta }_{ODD}\left(G\right)$ where ODD = 1,3,5,....

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...