Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {?1, 1} be a two-valued function. If k ≥ 1 is an integer and ${\sum_{x\in N[v]} f(x) \ge k}$ for each ${v \in V(G)}$ , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f ₁,f ₂, . . . ,f _d } of distinct signed k-dominating functions on G with the property that ${\sum_{i=1}^d f_i(x) \le k}$ for each ${x \in V(G)}$ , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.     

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ? that assigns to each element z ∈ V (G)∪E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ∑ _e∈E(u) ?(e)+?(u)≠∑ _e∈E(v) ?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ∈ L(z) for each element z ∈ V (G)∪E(G).     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Neighbor sum distinguishing total chromatic number of <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-minor free graph

A k-total coloring of a graph G is a mapping ?: V (G) ? E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ? E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ? is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ _Σ ^″ (G) the smallest value k in such a coloring of G: Pil?niak and Wo?niak conjectured that for any simple graph with maximum degree Δ(G), χ _Σ ^″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ _Σ ^″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ _Σ ^″ (G) = Δ(G) + 2.     

Signed total (k, k)-domatic number of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f: V(D) → {?1, 1} be a two-valued function. If k ≥?1 is an integer and ${\sum_{x \in N^-(v)}f(x) \ge k}$ for each ${v \in V(G)}$ , where N ^?(v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f ₁, f ₂, . . . , f _d } of signed total k-dominating functions on D with the property that ${\sum_{i=1}^df_i(x)\le k}$ for each ${x \in V(D)}$ , is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by ${d_{st}^{k}(D)}$ . In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on ${d_{st}^{k}(D)}$ . Some of our results are extensions of known properties of the signed total domatic number ${d_{st}(D)=d_{st}^{1}(D)}$ of digraphs D as well as the signed total domatic number d _st (G) of graphs G, given by Henning (Ars Combin. 79:277–288, 2006).     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

On trees with total domination number equal to edge-vertex domination number plus one

An edge e∈E(G) dominates a vertex v∈V(G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edge-vertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D?V(G) is a total dominating set of G if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. We characterize all trees with total domination number equal to edge-vertex domination number plus one.     

(3, 1)*-Choosability of graphs of nonnegative characteristic without intersecting short cycles

A graph G is called (k,d)^?-choosable if for every list assignment L satisfying ∣L(v)∣ ≥k for all v∈V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting i-cycles for all i=3,4,5 is (3,1)^?-choosable.     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

Minimum Difference Representations of Graphs

Define a k-minimum-difference-representation (k-MDR) of a graph G to be a family of sets \({\{S(v): v\in V(G)\}}\) such that u and v are adjacent in G if and only if min{|S(u)?S(v)|, |S(v)?S(u)|} ≥ k. Define ρ _min(G) to be the smallest k for which G has a k-MDR. In this note, we show that {ρ _min(G)} is unbounded. In particular, we prove that for every k there is an n ₀ such that for n > n ₀ ‘almost all’ graphs of order n satisfy ρ _min(G) > k. As our main tool, we prove a Ramsey-type result on traces of hypergraphs.     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.     

Diameter-Preserving Spanning Trees in Sparse Weighted Graphs

We give a simple sufficient condition for a weighted graph to have a diameter-preserving spanning tree. More precisely, let G = (V, E, f _E ) be a connected edge weighted graph with f _E being the edge weight function. Let f _V be the vertex weight function of G induced by f _E as follows: f _V (v) = max{f _E (e) : e is incident with v} for all \({v \in V}\) . We show that G contains a diameter-preserving spanning tree if \({d(G)\ge \frac{2}{3} \sum_{v\in V} f_V(v)}\) where d(G) is the diameter of G. The condition is sharp in the sense that for any \({\epsilon >0 }\) , there exist weighted graphs G satisfying \({d(G) > (\frac{2}{3}-\epsilon)\sum_{v\in V} f_V(v)}\) and not containing a diameter-preserving spanning tree.     

Lower negative decision number in a graph

Let G=(V,E) be a graph. A function f:V(G)→{?1,1} is called bad if ∑ _v∈N(v) f(v)≤1 for every v∈V(G). A bad function f of a graph G is maximal if there exists no bad function g such that g≠f and g(v)≥f(v) for every v∈V. The minimum of the values of ∑ _v∈V f(v), taken over all maximal bad functions f, is called the lower negative decision number and is denoted by β _D ^* (G). In this paper, we present sharp lower bounds on this number for regular graphs and nearly regular graphs, and we also characterize the graphs attaining those bounds.     

On signed distance-k-domination in graphs

The signed distance-k-domination number of a graph is a certain variant of the signed domination number. If v is a vertex of a graph G, the open k-neighborhood of v, denoted by N _k (v), is the set N _k (v) = {u: u ≠ v and d(u, v) ⩽ k}. N _k [v] = N _k (v) ⋃ {v} is the closed k-neighborhood of v. A function f: V → {−1, 1} is a signed distance-k-dominating function of G, if for every vertex . The signed distance-k-domination number, denoted by γ _k,s (G), is the minimum weight of a signed distance-k-dominating function on G. The values of γ _2,s (G) are found for graphs with small diameter, paths, circuits. At the end it is proved that γ _2,s (T) is not bounded from below in general for any tree T.     

Total [1,2]-domination in Graphs

A subset S ? V in a graph G = (V,E) is a total [1, 2]-set if, for every vertex \( \upsilon \in V, 1 \leq\mid N (\upsilon)\cap S\mid\leq \). The minimum cardinality of a total [1, 2]-set of G is called the total [1, 2]-domination number, denoted by γ_t[1,2](^G).We establish two sharp upper bounds on the total [1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extremal graphs achieving these bounds. Moreover, we give some sufficient conditions for a graph without total [1, 2]-set and for a graph with the same total [1, 2]-domination number, [1, 2]-domination number and domination number.     

On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

Some Existence Theorems on All Fractional (<Emphasis Type="Italic">g</Emphasis>, <Emphasis Type="Italic">f</Emphasis>)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) → Z⁺ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F _h such that E(H) ∩ E(F _h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

Some sufficient conditions for Hamiltonian property in terms of Wiener-type invariants

The Wiener-type invariants of a simple connected graph G = (V, E) can be expressed in terms of the quantities \(W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))\) for various choices of the function f(x), where d_G(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) 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 λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.