A Ramsey-type theorem for traceable graphs

A graph G is traceable if there is a path passing through all the vertices of G. It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A, illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences. It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K_ω,ω. The versions of these results for finite graphs are discussed.     

Matchings in 3-vertex-critical graphs: The odd case

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)<γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K_1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K_1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.     

Recursive generation of partitionable graphs

Results of Lovász (1972) and Padberg (1974) imply that partitionable graphs contain all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture. A recursive method of generating partitionable graphs was suggested by Chvátal, Graham, Perold, and Whitesides (1979). Results of Seb? (1996) entail that Berge's conjecture holds for all the partitionable graphs obtained by this method. Here we suggest a more general recursion. Computer experiments show that it generates all the partitionable graphs with ω=3,α ≤ 9 (and we conjecture that the same will hold for bigger α, too) and many but not all for (ω,α)=(4,4) and (4,5). Here, α and ω are respectively the clique and stability numbers of a partitionable graph, that is the numbers of vertices in its maximum cliques and stable sets. All the partitionable graphs generated by our method contain a critical ω‐clique, that is an ω‐clique which intersects only 2ω?2 other ω‐cliques. This property might imply that in our class there are no counterexamples to Berge's conjecture (cf. Seb? (1996)), however this question is still open. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 259–285, 2002     

Clique graphs of time graphs

A simple, finite graph G is called a time graph (equivalently, an indifference graph) if there is an injective real function f on the vertices v(G) such that v_iv_j ∈ e(G) for v_i ≠ v_j if and only if |f(v_i) ? f(v_j)| ≤ 1. A clique of a graph G is a maximal complete subgraph of G. The clique graph K(G) of a graph G is the intersection graph of the cliques of G. It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K²(G), and inductively, denote K(K^m?1(G)) by K^m(G). Define the index indx(G) of a connected time graph G as the smallest integer n such that Kⁿ(G) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.     

Finding minimum dominating cycles in permutation graphs

A dominating cycle for a graph G = (V, E) is a subset C of V which has the following properties: (i) the subgraph of G induced by C has a Hamiltonian cycle, and (ii) every vertex of V is adjacent to some vertex of C. In this paper, we develop an O(n²) algorithm for finding a minimum cardinality dominating cycle in a permutation graph. We also show that a minimum cardinality dominating cycle in a permutation graph always has an even number of vertices unless it is isomorphic to C₃.     

Clique-perfectness of complements of line graphs

The clique-transversal number τ_c(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number α_c(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph complementation nor is a characterization by forbidden induced subgraphs known. Nevertheless, partial results in this direction have been obtained. For instance, in [Bonomo, F., M. Chudnovsky and G. Durán, Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs, Discrete Appl. Math. 156 (2008), pp. 1058–1082], a characterization of those line graphs that are clique-perfect is given in terms of minimal forbidden induced subgraphs. Our main result is a characterization of those complements of line graphs that are clique-perfect, also by means of minimal forbidden induced subgraphs. This implies an O(n²) time algorithm for deciding the clique-perfectness of complements of line graphs and, for those that are clique-perfect, finding α_c and τ_c.     

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω_c(G-x)>ω(G-x) for some x∈V(G), then G=K_(2k+1)/2+x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.     

Cycle-magic graphs

A simple graph G=(V,E) admits a cycle-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given cycle C. Then the graph G is C-magic if there exists a total labelling f:V∪E→{1,2,…,|V|+|E|} such that, for every subgraph H^′=(V^′,E^′) of G isomorphic to C, ∑_v∈V^′f(v)+∑_e∈E^′f(e) is constant. When f(V)={1,…,|V|}, then G is said to be C-supermagic.We study the cyclic-magic and cyclic-supermagic behavior of several classes of connected graphs. We give several families of C_r-magic graphs for each r?3. The results rely on a technique of partitioning sets of integers with special properties.     

On algebraic connectivity augmentation

Suppose that G is an undirected graph, and that H is a spanning subgraph of G^c whose edges induce a subgraph on p vertices. We consider the expression α(G∪H)-α(G), where α denotes the algebraic connectivity. Specifically, we provide upper and lower bounds on α(G∪H)-α(G) in terms of p, and characterise the corresponding equality cases. We also discuss the density of the expression α(G∪H)-α(G) in the interval [0,p]. A bound on α(G∪H)-α(G) is provided in a special case, and several examples are considered.     

Path factors and parallel knock-out schemes of almost claw-free graphs

An H₁,{H₂}-factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H₁ and all other components (if there are any) isomorphic to the graph H₂. We completely characterise the class of connected almost claw-free graphs that have a P₇,{P₂}-factor, where P₇ and P₂ denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation, we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost claw-free graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n^5.376) time algorithm for claw-free graphs on n vertices).     

Eigenvalues of a graph and its imbeddings

A graph H is imbedded in a graph G if a subset of the vertices of G determines a subgraph isomorphic to H. If λ(G) is the least eigenvalue of G and k_R(H) = lim sup_d→∞ {λ(G)| H imbedded in G; G regular and connected; diam(G) > d; deg(G) > d}, then λ(H) ? 2 ≤ k_R(H) ≤ λ(H) with these bounds being the best possible. Given a graph H, there exist arbitrarily large families of isospectral graphs such that H can be imbedded in each member of the family.     

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1,2]-factor F_G, i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α(G)=α(F_G) for some perfect [1,2]-factor F_G. This class contains all well-covered graphs G without isolated vertices of order n with α?(n-1)/2, and in particular all very well-covered graphs.     

The chromatic difference sequence of a graph

Let α_k(G) denote the maximum number of vertices in a k-colorable subgraph of G. Set α_k(G)=α_k(G)?α_(k?1)(G). The sequence a₁(G), a₂(G),… is called the chromatic difference sequence of the graph G. We present necessary and sufficient conditions for a sequence to be the chromatic difference sequence of some 4-colorable graph.     

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

A matching M is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139-154]. G is a König-Egerváry graph provided α(G)+μ(G)=|V(G)| [R.W. Deming, Independence numbers of graphs—an extension of the König-Egerváry theorem, Discrete Math. 27 (1979) 23-33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228-229], where μ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, and we write S∈Ψ(G), if S is a maximum stable set of the subgraph spanned by S∪N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232-248], proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163-174] we have proved that for a bipartite graph G,Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G is a triangle-free graph, then Ψ(G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S∈Ψ(G), the subgraph spanned by S∪N(S) is a König-Egerváry graph.     

On self-clique graphs with given clique sizes, II

The clique graph of a graph G is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques have a non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. We give a new characterization of the set of all connected self-clique graphs having all cliques but two of size 2.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

On 3-Edge-Connected Supereulerian Graphs

The supereulerian graph problem, raised by Boesch et al. (J Graph Theory 1:79–84, 1977), asks when a graph has a spanning eulerian subgraph. Pulleyblank showed that such a decision problem, even when restricted to planar graphs, is NP-complete. Jaeger and Catlin independently showed that every 4-edge-connected graph has a spanning eulerian subgraph. In 1992, Zhan showed that every 3-edge-connected, essentially 7-edge-connected graph has a spanning eulerian subgraph. It was conjectured in 1995 that every 3-edge-connected, essentially 5-edge-connected graph has a spanning eulerian subgraph. In this paper, we show that if G is a 3-edge-connected, essentially 4-edge-connected graph and if for every pair of adjacent vertices u and v, d _G (u) + d _G (v) ≥ 9, then G has a spanning eulerian subgraph.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

On a general class of graph polynomials

Let $\text{F}$ be a family of connected graphs. With each element α ∈ $\text{F}$, we can associate a weight w_α. Let G be a graph. An F-cover of G is a spanning subgraph of G in which every component belongs to $\text{F}$. With every F-cover we can associate a monomial π(C) = Π_αw_α, where the product is taken over all components of the cover. The F-polynomial of G is Σπ(C), where the sum is taken over all F-covers in G. We obtain general results for the complete graph and complete bipartite graphs, and we show that many of the well-known graph polynomials are special cases of more general F-polynomials.