Geodesic subgraphs

Define a geodesic subgraph of a graph to be a subgraph H with the property that any geodesic of two points of H is in H. The trivial geodesic subgraphs are the complete graphs K_n' n ≧ 0, and G itself. We characterize all (finite, simple, connected) graphs with only the trivial geodesic subgraphs, and give an algorithm for their construction. We do this also for triangle-free graphs.     

On end degrees and infinite cycles in locally finite graphs

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.     

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     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

Lambda composition

A lambda in a graph G is two edges uv and vw such that uw is not an edge. A subgraph A of G is called a lambda-subgraph if every lambda of G has both or neither of its edges in A. We describe the decomposition of a graph into its lambda subgraphs and use this to prove a decomposition theorem of Gallai (Acta Math. Acad. Sci. Hungar. 18 (1967), 25–66). A corollary is that a graph is perfect if and only if each of its edge-minimal lambda subgraphs is. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:9–16, 1997     

Matroidal families of finite connected nonhomeomorphic graphs exist

A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.     

Small Edge Sets Meeting all Triangles of a Graph

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K ₄-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.     

Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let Forb*(H) be the class of graphs with no induced subdivision of H. It was conjectured in [J Graph Theory, 24 (1997), 297–311] that, for every graph H, there is a function f_H: ?→? such that for every graph G∈Forb*(H), χ(G)≤f_H(ω(G)). We prove this conjecture for several graphs H, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex‐disjoint pendant edges), and what we call a “necklace,” that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:49–68, 2012     

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186-209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253-267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown.We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time.Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.     

The Linear Arboricity of Series-Parallel Graphs

 The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. A graph is called series-parallel if it contains no subgraphs homeomorphic to K ₄. In this paper, we prove that for any series-parallel graph G having Δ (G)≥3. Since an outerplanar graph is a series-parallel graph, this is also true for any outerplanar graph. Received: August 20, 1997 Revised: March 12, 1999     

Contractible elements in k‐connected graphs not containing some specified graphs

In 15 , Thomassen proved that any triangle‐free k‐connected graph has a contractible edge. Starting with this result, there are several results concerning the existence of contractible elements in k‐connected graphs which do not contain specified subgraphs. These results extend Thomassen's result, cf., 2 , 3 , 9 - 13 . In particular, Kawarabayashi 12 proved that any k‐connected graph without K subgraphs contains either a contractible edge or a contractible triangle. In this article, we further extend these results, and prove the following result. Let k be an integer with k ≥ 6. If G is a k‐connected graph such that G does not contain as a subgraph and G does not contain as an induced subgraph, then G has either a contractible edge which is not contained in any triangle or a contractible triangle. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:97–109, 2008     

Contractible Cliques in k-Connected Graphs

Kawarabayashi proved that for any integer k≥4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen's result that every triangle-free k-connected graph contains a k-contractible edge. In this paper, we extend Kawarabayashi's technique and prove a more general result concerning k-contractible cliques. Xingxing Yu was partially supported by NSF grant DMS-0245530 and NSA grant MDA-904-03-1-0052.     

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H, for every induced subgraph H of G. A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W₄, bull}-free, both superclasses of triangle-free graphs.     

A graph-theoretic version of the union-closed sets conjecture

An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e occurs in more than half of the derived subgraphs of G. We introduce the conjecture: Every non-empty graph contains a non-residual edge. This conjecture is implied by, but weaker than, the union-closed sets conjecture. We prove that a graph G of order n satisfies this conjecture whenever G satisfies any one of the conditions: δ(G) ≤ 2, log₂ n ≤ δ(G), n ≤ 10, or the girth of G is at least 6. Finally, we show that the union-closed sets conjecture, in its full generality, is equivalent to a similar conjecture about hypergraphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 155–163, 1997     

The fractional chromatic number of infinite graphs

The fractional chromatic number of a graph G is the infimum of the total weight that can be assigned to the independent sets of G in such a way that, for each vertex v of G, the sum of the weights of the independent sets containing v is at least 1. In this note we give a graph a graph whose fractional chromatic number is strictly greater than the supremum of the fractional chromatic numbers of its finite subgraphs. This answers a question of Zhu. We also give some grphs for which the fractional chromatic number is not attined, answering another of Zhu. © 1995 John Wiley & Sons, Inc.     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

The poset on connected induced subgraphs of a graph need not be Sperner

SupposeG is a finite connected graph. LetC(G) denote the inclusion ordering on the connected vertex-induced subgraphs ofG. Penrice asked whetherC(G) is Sperner for general graphsG. Answering Penrice's question in the negative, we present a treeT such thatC(T) is not Sperner. We also construct a related distributive lattice that is not Sperner.     

How to decrease the diameter of triangle-free graphs

Assume that G is a triangle-free graph. Let be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that for connected graphs of order n and of fixed maximum degree. The proof is based on relations of and the clique-cover number of edges of graphs. It is also shown that the maximum value of over (triangle-free) graphs of order n is . The behavior of is different, its maximum value is . We could not decide whether for connected (triangle-free) graphs of order n with a positive ε. Received: October 12, 1997