Product irregularity strength of graphs

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w:E(G)→{1,2,…,m} is called product-irregular, if all product degrees pd_G(v)=∏_e∋vw(e) are distinct. The goal is to obtain a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength. The analogous concept of irregularity strength, with sums in place of products, has been introduced by Chartrand et al. and investigated by many authors.     

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.     

On large semi-linked graphs

Let H$H$ be a multigraph, possibly with loops, and consider a set S⊆V(H)$S\subseteq V\left(H\right)$. A (simple) graph G$G$ is (H,S)$(H,S)$-semi-linked   if, for every injective map f:S→V(G)$f:S\to V\left(G\right)$, there exists an injective map g:V(H)?S→V(G)?f(S)$g:V\left(H\right)?S\to V\left(G\right)?f\left(S\right)$ and a set of |E(H)|$\left|E\left(H\right)\right|$ internally disjoint paths in G$G$ connecting pairs of vertices of  f(S)∪g(V(H)?S)$f\left(S\right)\cup g(V\left(H\right)?S)$ for every edge between the corresponding vertices of H$H$. This new concept of (H,S)$(H,S)$-semi-linkedness is a generalization of H$H$-linkedness  . We establish a sharp minimum degree condition for a sufficiently large graph G$G$ to be (H,S)$(H,S)$-semi-linked.     

On λ-harmonic graphs

A λ harmonic graph G, a λ-Hgraph G for short, means that there exists a constant denotes the degree of vertex vi. In this paper, some harmonic properties of the complement and line graph are given, and some algebraic properties for the λ-Hgraphs are obtained.     

On subpancyclic line graphs

We give a best possible Dirac-like condition for a graph G so that its line graph L(G) is subpancyclic, i.e., L(G) contains a cycle of length l for each l between 3 and the circumference of G. The result verifies the conjecture posed by Xiong (Pancyclic results in hamiltonian line graphs, in: Combinatorics and Graph Theory '95, vol. 2, Proceedings of the Summer School and International Conference on Combinatorics, World Scientific). © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 67–74, 1998     

ContributionsIrregularity strength of trees

The main result of this paper establishes that the irregularity strength of any tree with no vertices of degree two is its number of pendant vertices.     

Irregularity strength of trees

The main result of this paper establishes that the irregularity strength of any tree with no vertices of degree two is its number of pendant vertices.     

On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

On decomposition of triangle-free graphs under degree constraints

We prove that if s and t are positive integers and if G is a triangle-free graph with minimum degree s + t, then the vertex set of G has a decomposition into two sets which induce subgraphs of minimum degree at least s and t, respectively. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 7–9, 1998     

Edge irregular total labellings for graphs of linear size

As an edge variant of the well-known irregularity strength of a graph G=(V,E) we investigate edge irregular total labellings, i.e. functions f:V∪E→{1,2,…,k} such that f(u)+f(uv)+f(v)≠f(u^′)+f(u^′v^′)+f(v^′) for every pair of different edges uv,u^′v^′∈E. The smallest possible k is the total edge irregularity strength of G. Confirming a conjecture by Ivan?o and Jendrol’ for a large class of graphs we prove that the natural lower bound is tight for every graph of order n, size m and maximum degree Δ with m>111000Δ. This also implies that the probability that a random graph from G(n,p(n)) satisfies the Ivan?o-Jendrol’ Conjecture tends to 1 as n→∞ for all functions p∈[0,1]^N. Furthermore, we prove that is an upper bound for every graph G of order n and size m≥3 whose edges are not all incident to a single vertex.     

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.     

关于图的最小Q-特征值

研究了基于n阶二部图和s阶完全图构造的一个图类,得到了该图类的无符号拉普拉斯最小特征值(即最小Q-特征值)的一个可达上界为s.基于此,对于任意给定的正整数s和正偶数n,构造了最小Q-特征值为s的一类n+s阶图.另外,对于任意给定的最小度δ和阶数n,在满足2≤δ≤n-1/2条件下,构造了最小Q-特征值为δ-1的一类n阶图.     

On k‐ordered graphs

Ng and Schultz [J Graph Theory 1 ( 6 ), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k‐ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k‐ordered Hamiltonian. We give sum of degree conditions for nonadjacent vertices and neighborhood union conditions that imply a graph is k‐ordered Hamiltonian. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 69–82, 2000     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate [A.A. Ageev, Every circle graph with girth at least 5 is 3-colourable, Discrete Math. 195 (1999) 229-233]. In this paper, we prove that circle graphs with girth at least g≥5 and minimum degree at least two contain a chain of g−4 vertices of degree two, which implies Ageev’s result in the case g=5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g≥5 as well as a precise estimate of their maximum average degree.