Every graph occurs as an induced subgraph of some hypohamiltonian graph

We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K₃, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph.     

Barycentric systems and stretchability

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible.     

Preperfect graphs

We say that a vertexx of a graph is predominant if there exists another vertexy ofG such that either every maximum clique ofG containingy containsx or every maximum stable set containingx containsy. A graph is then called preperfect if every induced subgraph has a predominant vertex. We show that preperfect graphs are perfect, and that several well-known classes of perfect graphs are preperfect. We also derive a new characterization of perfect graphs.     

赋权图的w密度和w平衡性

The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced. A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it. In this paper,a good characterization of w-balanced weighted graphs is given. Applying this characterization ,many large w-balanced weighted graphs are formed by combining smaller ones. In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed. It is shown that the w-density theory is closely related to the study of SEW(G,w) games.     

Connected graphs containing a given connected graph as a unique greatest common subgraph

Those connected graphsG are determined for which there exist nonisomorphic connected graphs of equal size containingG as a unique greatest common subgraph. Analogous results are also obtained for weakly connected and strongly connected digraphs, as well as for induced subgraphs and induced subdigraphs.This research was supported by a Western Michigan University faculty research fellowship.This research was supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.     

Hereditary modular graphs

In a hereditary modular graphG, for any three verticesu, v, w of an isometric subgraph ofG, there exists a vertex of this subgraph that is simultaneously on some shortestu, v-path,u, w-path andv, w-path. It is shown that the hereditary modular graphs are precisely those bipartite graphs which do not contain any isometric cycle of length greater than four. There is a polynomial-time algorithm available which decides whether a given (bipartite) graph is hereditary modular or not. Finally, the chordal bipartite graphs are characterized by forbidden isometric subgraphs.     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

Global cycle properties in graphs with large minimum clustering coefficient

Let 𝒫 be a graph property. A graph G is said to be locally 𝒫 (closed locally 𝒫) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property 𝒫. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficient among all vertices of G. Let H be a subgraph of a graph G and let S ? V (H). We say that H is a strongly induced subgraph of G with attachment set S, if H is an induced subgraph of G and the vertices of V (H) ? S are not incident with edges that are not in H. A graph G is fully cycle extendable if every vertex of G lies in a triangle and for every nonhamiltonian cycle C of G, there is a cycle of length |V (C)|?+?1 that contains the vertices of C. A complete characterization, of those locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 that are fully cycle extendable, is given in terms of forbidden strongly induced subgraphs (with specified attachment sets). Moreover, it is shown that all locally connected graphs with Δ?≤?6 and sufficiently large minimum clustering coefficient are weakly pancylic, thereby proving Ryj´ǎcek’s conjecture for this class of graphs.     

The complexity of a class of infinite graphs

IfG _k is the family of countable graphs with nok vertex (or edge) disjoint circuits (1<k<) then there is a countableG _k G _k such that every member ofG _k is an (induced) subgraph of some member ofG _k , but no finiteG _k suffices.     

Connected Domination Stable Graphs Upon Edge Addition

Abstract

A set S of vertices in a graph G is a connected dominating set of G if S dominates G and the subgraph induced by S is connected. We study the graphs for which adding any edge does not change the connected domination number.     

Alternating cycle-free matchings

The investigation of alternating cycle-free matchings is motivated by the Jump-number problem for partially ordered sets and the problem of counting maximum cardinality matchings in hexagonal systems.We show that the problem of deciding whether a given chordal bipartite graph has an alternating cycle-free matching of a given cardinality is NP-complete. A weaker result, for bipartite graphs only, has been known for some time. Also, the alternating cycle-free matching problem remains NP-complete for strongly chordal split graphs of diameter 2.In contrast, we give algorithms to solve the alternating cycle-free matching problem in polynomial time for bipartite distance hereditary graphs (time O(m ²) on graphs with m edges) and distance hereditary graphs (time O(m ⁵)).     

Minimum cycle bases of graphs on surfaces

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1-1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total length. This provides a way to enumerate MCBs in a graph via nonseparating cycles. In particular, some known results such as P.F. Stadler's work on Halin graphs [Minimum cycle bases of Halin graphs, J. Graph Theory 43 (2003) 150-155] and Leydold and Stadler's results on outer-planar graphs [Minimum cycle bases of outerplanar graphs, Electronic J. Combin. 5(16) (1998) 14] are concluded. As applications, the number of MCBs in some types of graphs embedded in lower surfaces (with arbitrarily high genera) is found. Finally, we present an interpolation theorem for the number of one-sided cycles contained in MCB of an embedded graph.     

A new bound for neighbor-connectivity of abelian Cayley graphs

For the notion of neighbor-connectivity in graphs, whenever a vertex is “subverted” the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is bounded above by κ. The main result of this paper is a sharpening of the bound for abelian Cayley graphs. In particular, we show by constructing an effective subversion strategy for such graphs, that neighbor-connectivity is bounded above by ⌈δ/2⌉+2. Using a result of Watkins the new bound can be recast in terms of κ to get neighbor-connectivity bounded above by ⌈3κ/4⌉+2 for abelian Cayley graphs.     

On Triangle‐Free Graphs That Do Not Contain a Subdivision of the Complete Graph on Four Vertices as an Induced Subgraph

We prove a decomposition theorem for the class of triangle‐free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least five in this class is 3‐colorable.     

Graph colourings with forbidden k-coloured subgraphs

Abstract

For a set F of graphs and a natural number k, an (F, k)-colouring of a graph G is a proper colouring of V (G) such that no subgraph of G isomorphic to an element of F is coloured with at most k colours. Equivalently, if P is the class of all graphs that do not contain an element of F as a subgraph, a χP,k colouring of G is a proper colouring such that the union of at most k colour classes induces a graph in P. The smallest number of colours in such a colouring of G, if it exists, is denoted by χP,k (G). We give some general results on χP,k-colourings and investigate values of χP,k (G) for some choices of P and classes of graphs G.     

The asymptotic number of graphs not containing a fixed color-critical subgraph

For a finite graphG letForb(H) denote the class of all finite graphs which do not containH as a (weak) subgraph. In this paper we characterize the class of those graphsH which have the property that almost all graphs inForb(H) are -colorable. We show that this class corresponds exactly to the class of graphs whose extremal graph is the Turán-graphT _n ().An earlier result of Simonovits (Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions,Discrete Math. 7 (1974), 349–376) shows that these are exactly the (+1)-chromatic graphs which contain a color-critical edge.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.     

Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs

Recently much attention has been focused on the theory of quasi-random graph and hypergraph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. We shall investigate propertiesP which do not imply quasi-randomnes for sequences (G _n ) of graphs on their own, but do imply if they hold not only for the whole graphG _n but also for every sufficiently large subgraph ofG _n . Here the properties are strongly connected to countingnot necessarily induced subgraphs of a given type, while in a subsequent paper we shall investigate the properties connected with counting induced subgraphs.Dedicated to the memory of Paul ErdsResearch supported by OTKA N1909.