Edge colouring by total labellings

We introduce the concept of an edge-colouring total k-labelling. This is a labelling of the vertices and the edges of a graph G with labels 1,2,…,k such that the weights of the edges define a proper edge colouring of G. Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define to be the smallest integer k for which G has an edge-colouring total k-labelling. This parameter has natural upper and lower bounds in terms of the maximum degree Δ of . We improve the upper bound by 1 for every graph and prove . Moreover, we investigate some special classes of graphs.     

On the spanning fan-connectivity of graphs

Let G be a graph. The connectivity of G, κ(G), is the maximum integer k such that there exists a k-container between any two different vertices. A k-container of G between u and v, C_k(u,v), is a set of k-internally-disjoint paths between u and v. A spanning container is a container that spans V(G). A graph G is k^∗-connected if there exists a spanning k-container between any two different vertices. The spanning connectivity of G, κ^∗(G), is the maximum integer k such that G is w^∗-connected for 1≤w≤k if G is 1^∗-connected.Let x be a vertex in G and let U={y₁,y₂,…,y_k} be a subset of V(G) where x is not in U. A spanningk−(x,U)-fan, F_k(x,U), is a set of internally-disjoint paths {P₁,P₂,…,P_k} such that P_i is a path connecting x to y_i for 1≤i≤k and . A graph G is k^∗-fan-connected (or -connected) if there exists a spanning F_k(x,U)-fan for every choice of x and U with |U|=k and x∉U. The spanning fan-connectivity of a graph G, , is defined as the largest integer k such that G is -connected for 1≤w≤k if G is -connected.In this paper, some relationship between κ(G), κ^∗(G), and are discussed. Moreover, some sufficient conditions for a graph to be -connected are presented. Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be k^∗-pipeline-connected.     

Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k≥0, the intersection graph Q_k(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H_x and H_y in Q_k(G) are adjacent whenever the intersection H_x∩H_y contains a subgraph isomorphic to Q_k. Characterizations of clique-graphs in terms of these intersection concepts when k>0, are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted , whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.     

Dimension-2 poset competition numbers and dimension-2 poset double competition numbers

Let D=(V(D),A(D)) be a digraph. The competition graph of D, is the graph with vertex set V(D) and edge set . The double competition graph of D, is the graph with vertex set V(D) and edge set . A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R² and there is an arc going from a vertex (x₁,y₁) to a vertex (x₂,y₂) if and only if x₁>x₂ and y₁>y₂. We show that a graph is the competition graph of a poset of dimension at most two if and only if it is an interval graph, at least half of whose maximal cliques are isolated vertices. This answers an open question on the doubly partial order competition number posed by Cho and Kim. We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing a result of Kim, Kim, and Rho. Some connections are also established between the minimum numbers of isolated vertices required to be added to change a given graph into the competition graph, the double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.     

Small alliances in a weighted graph

We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑_{x∈NG(v)∩S}w(x)+w(v)≥∑_{x∈NG(v)−S}w(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3].     

Total edge irregularity strength of complete graphs and complete bipartite graphs

A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs.     

Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

A result on the total colouring of powers of cycles

The total chromatic number χ_T(G) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that for every simple graph G, χ_T(G)?Δ(G)+2. This work verifies the TCC for powers of cycles even and 2<k<n/2, showing that there exists and can be polynomially constructed a (Δ(G)+2)-total colouring for these graphs.     

A note on edge-graceful spectra of the square of paths

For a simple path P_r on r vertices, the square of P_r is the graph on the same set of vertices of P_r, and where every pair of vertices of distance two or less in P_r is connected by an edge. Given a (p,q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k,k+1,…,k+q−1, so that the induced vertex sums are pairwise distinct, where the vertex sum at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices p. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI(G). In this article, the edge-graceful spectrum for the square of paths is completely determined for odd r.     

The 6-relaxed game chromatic number of outerplanar graphs

Suppose G is a graph and k,d are integers. The (k,d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice’s goal is to have all the vertices coloured, and Bob’s goal is the opposite: to have an uncoloured vertex without a legal colour. The d-relaxed game chromatic number of G, denoted by , is the least number k so that when playing the (k,d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then for d≥6.     

A new upper bound for the total vertex irregularity strength of graphs

We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting w of a graph G=(V,E) with elements of a set {1,2,…,s}, denote wt_G(v)=∑_evw(e)+w(v) for each vV. The smallest s for which exists such a weighting with wt_G(u)≠wt_G(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by . We prove that for each graph of order n and with minimum degree δ>0.     

On maximum planar induced subgraphs

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G^′. In this case G^′ is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)?k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)?k. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.     

Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of 2n-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271-277] proved that for even integer k, if G-x has a k-factor for each x∈V(G), then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239-242] generalized Katerinis’ result to f-factors, and proved that if G-x has an f-factor for each x∈V(G), then G has an f-factor for an integer-valued function f defined on V(G) with even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether G-x has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on V(G) with even, |V(G)-X|?2. We prove that if and if G-x has an f-factor for each x∈V(G)-X, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition with . Furthermore the condition will be when |X|=1.     

Graphs with maximal signless Laplacian spectral radius

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δ_r>δ_r-1>?>δ₁, it is proved that ρ(Q(G))<ρ(Q(H)) for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer such that δ_i+δ_r+1-i?n+1 (respectively, δ_i+δ_r+1-i?δ_l+δ_r-l+1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m-k vertices, for k=0,1,2,3, are completely determined.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.