Distance-residual subgraphs

For a connected finite graph G and a subset V₀ of its vertex set, a distance-residual subgraph is a subgraph induced on the set of vertices at the maximal distance from V₀. Some properties and examples of distance-residual subgraphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are presented. The relations between the distance-residual subgraphs of product graphs and their factors are explored.     

Algebraic connectivity for vertex-deleted subgraphs, and a notion of vertex centrality

Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G?v. Denote the algebraic connectivities of G and G?v by α(G) and α(G?v), respectively. In this paper, we consider the functions ?(v)=α(G)−α(G?v) and , provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function κ yields a measure of vertex centrality, and we apply that measure to analyse certain graphs arising from food webs.     

Conditions for graphs to be path partition optimal

The path partition number of a graph is the minimum number of edges we have to add to turn it into a Hamiltonian graph, and the separable degree is the minimum number of edges we have to add to turn it into a 2-connected graph. A graph is called path partition optimal if its path partition number is equal to its separable degree. We study conditions that guarantee path partition optimality. We extend several known results on Hamiltonicity to path partition optimality, in particular results involving degree conditions and induced subgraph conditions.     

Mock threshold graphs

Mock threshold graphs are a simple generalization of threshold graphs that, like threshold graphs, are perfect graphs. Our main theorem is a characterization of mock threshold graphs by forbidden induced subgraphs. Other theorems characterize mock threshold graphs that are claw-free and that are line graphs. We also discuss relations with chordality and well-quasi-ordering as well as algorithmic aspects.     

Subgraph‐avoiding coloring of graphs

Given a “forbidden graph” F and an integer k, an F‐avoiding k‐coloring of a graph G is a k‐coloring of the vertices of G such that no maximal F‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac_F(G) is the smallest integer k such that G is F‐avoiding k‐colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that ac_F(G) ? C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac_F(G) in terms of various invariants of G are also given. Particularly, we prove that ${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}Given a “forbidden graph” F and an integer k, an F‐avoiding k‐coloring of a graph G is a k‐coloring of the vertices of G such that no maximal F‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac_F(G) is the smallest integer k such that G is F‐avoiding k‐colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that ac_F(G) ? C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac_F(G) in terms of various invariants of G are also given. Particularly, we prove that ${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}$, where n is the order of G and F is not K_k or $\overline{{{K}}_{{{k}}}}$. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300–310, 2010     

Light subgraphs in planar graphs of minimum degree 4 and edge‐degree 9

A graph H is light in a given class of graphs if there is a constant w such that every graph of the class which has a subgraph isomorphic to H also has a subgraph isomorphic to H whose sum of degrees in G is ≤ w. Let be the class of simple planar graphs of minimum degree ≥ 4 in which no two vertices of degree 4 are adjacent. We denote the minimum such w by w(H). It is proved that the cycle C_s is light if and only if 3 ≤ s ≤ 6, where w(C₃) = 21 and w(C₄) ≤ 35. The 4‐cycle with one diagonal is not light in , but it is light in the subclass consisting of all triangulations. The star K_1,s is light if and only if s ≤ 4. In particular, w(K_1,3) = 23. The paths P_s are light for 1 ≤ s ≤ 6, and heavy for s ≥ 8. Moreover, w(P₃) = 17 and w(P₄) = 23. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 261–295, 2003     

Wiener Index of Hexagonal Systems

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.     

Directed cycles with chords

Using a variation of Thomassen's admissible triples technique, we give an alternative proof that every strongly 2‐arc‐connected directed graph with two or more vertices contains a directed cycle that has at least two chords, while at the same time establishing a more general result. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:17–28, 1999     

Forbidden ordered subgraph vs. forbidden subgraph characterizations of graph classes

We show that the minimum set of unordered graphs that must be forbidden to get the same graph class characterized by forbidding a single ordered graph is infinite. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 71–76, 1999