On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332-334] conjectured that if |V(G)|≡0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.     

Finite common coverings of pairs of regular graphs

The graph G is a covering of the graph H if there exists a (projection) map p from the vertex set of G to the vertex set of H which induces a one-to-one correspondence between the vertices adjacent to v in G and the vertices adjacent to p(v) in H, for every vertex v of G. We show that for any two finite regular graphs G and H of the same degree, there exists a finite graph K that is simultaneously a covering both of G and H. The proof uses only Hall's theorem on 1-factors in regular bipartite graphs.     

Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K^-1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.     

Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz

For a graph G, let χ(G) denote its chromatic number and σ(G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of χ(G)=σ(G) over all n-vertex graphs G. A famous conjecture of Hajós from 1961 states that σ(G) ≥ χ(G) for every graph G. That is, H(n)≤1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erd?s and Fajtlowicz further showed by considering a random graph that H(n)≥cn ^1/2/logn for some absolute constant c>0. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant C such that χ(G)=σ(G) ≤ Cn ^1/2/logn for all n-vertex graphs G. In this paper we prove the Erd?s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on n vertices with independence number α.     

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.     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.     

The clique operator on circular-arc graphs

A circular-arc graphG is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graphK(G) of a graph G is the intersection graph of its cliques. The iterated clique graphKⁱ(G) of G is defined by K⁰(G)=G and Kⁱ⁺¹(G)=K(Kⁱ(G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K-converges, if it is K-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.     

Edge covering coloring of nearly bipartite graphs

LetG be a simple graph with vertex setV(G) and edge setE(G). A subsetS ofE(G) is called an edge cover ofG if the subgraph induced byS is a spanning subgraph ofG. The maximum number of edge covers which form a partition ofE(G) is called edge covering chromatic number ofG, denoted by χ′_c(G). It known that for any graphG with minimum degreeδ,δ -1 ≤χ′_c(G) ≤δ. If χ′_c(G) =δ, thenG is called a graph of CI class, otherwiseG is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.     

Duality and perfection for edges in cliques

Given a graph G, let K(G) denote the graph whose vertices correspond with the edges of G. Two vertices of K(G) are joined by an edge if the corresponding edges in G are contained in a clique. This paper investigates some properties of G which force duality theorems for K(G).     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

A polynomial algorithm for the minimum weighted clique cover problem on claw-free perfect graphs

Let G = (V, E) be a graph with a positive number wt(v) assigned to each v ? v. A weighted clique cover of the vertices of G is a collection of cliques with a non-negative weight y_c assigned to each clique C in the collection such that σ_C:v?CY_C?wt(v) for all v ? V. The problem considered is to minimize σ_Cy_C over all weighted clique covers. A polynomial time algorithm for this problem is presented for graphs that are claw-free and perfect.     

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.     

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.     

The square of every two-connected graph is Hamiltonian

A graph G is a line-critical block if κ(G) = 2 and if for any line e of G the graph G ? e has κ(G ? e) = 1.If G is a line-critical block, then G is either a DT-block (i.e., G is a two-connected graph in which every line is incident to a point of degree two), or G contains a specific two-connected subgraph which is a DT-block (Theorem 1). Using this result and results of the preceding paper on DT-graphs, a simple proof of the conjecture that the square of every two-connected graph is Hamiltonian is given.     

Contractibility and the clique graph operator

To any graph G we can associate a simplicial complex Δ(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever Δ(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whose clique graph is a cone is contractible.     

Finding minimum clique capacity

Let C be a clique of a graph G. The capacity of C is defined to be (|V (G)\C|+|D|)/2, where D is the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C. We give a polynomial-time algorithm to find the minimum clique capacity in a graph G. This problem arose in the study [1] of packing vertex-disjoint induced three-vertex paths in a graph with no stable set of size three, which in turn was motivated by Hadwiger’s conjecture.     

A method in graph theory

A unified approach to a variety of graph-theoretic problems is introduced. The k-closure C_k(G) of a simple graph G of order n is the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree-sum at least k. It is shown that, for many properties P, one can find a suitable value of k (depending on P and n) such that if C_k(G) has P, then so does G. For instance, if P is the hamiltonian property, one may take k = n. Thus if C_n(G) is hamiltonian, then so is G; in particular, if n ? 3 and C_n(G) is complete, then G is hamiltonian. This condition for a graph to be hamiltonian is shown to imply the well-known conditions of Chvátal and Las Vergnas. The same method, applied to other properties, yields many new theorems of a similar nature.