Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs.     

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K₄, then . In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then unless G is isomorphic to K₄ (in which case its boxicity is 1).     

Vertex- and edge-minimal and locally minimal graphs

Given a family of graphs , a graph is called edge-minimal (vertex-minimal) if for every subgraph (induced subgraph) G^′ of G; furthermore, G is called locally edge-minimal (locally vertex-minimal) if whenever G^′ is obtained from G by deleting an edge (a vertex). Similarly, the concepts of minimality and local minimality are introduced for directed graphs (digraphs) and, more generally, for partially ordered sets.For example, by the Strong Perfect Graph Theorem, the only vertex-minimal graphs with χ>ω are odd holes and anti-holes. In contrast, the only locally vertex-minimal graphs with χ>ω are partitionable graphs. Somewhat surprisingly, there are infinitely many non-trivial perfect graphs that are locally edge-minimal and -maximal simultaneously. In other words, such a graph is perfect but it becomes imperfect after any edge is deleted from or added to it.In this paper we consider vertex- and edge-minimal and locally minimal graphs in the following families: (i) perfect and imperfect graphs, (ii) graphs with χ=ω and with χ>ω, (iii) digraphs that have a kernel and kernel-free digraphs, and finally, (iv) vertex-minimal complementary connected d-graphs.     

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.     

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

A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H 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 cliqueperfect 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, two superclasses of triangle-free graphs.     

The galaxies of nonstandard enlargements of infinite and transfinite graphs

The galaxies of the nonstandard enlargements of connected, conventionally infinite graphs as well as of walk-connected transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their “closeness” to the principal galaxy. If an enlargement has a galaxy Γ different from its principal galaxy, then it has a two-way infinite sequence of galaxies that contains Γ and is totally ordered according to that “closeness” property. There may be many such totally ordered sequences.Furthermore, a walk-connected graph G¹ of transfinite rank 1 consists in general of connected conventional graphs (graphs of rank 0, called 0-sections) that are walk-connected together at their infinite extremities. The enlargement of G¹ consists of the enlargement of G¹, as well as of the enlargements of its 0-sections. The latter enlargements are all contained within the principal galaxy of . Moreover, may have other galaxies of rank 1; these too are partially and totally ordered as before. These results extend to the enlargements of transfinite graphs of ranks greater than 1.     

On orthogonal ray graphs

An orthogonal ray graph is an intersection graph of horizontal and vertical rays (half-lines) in the xy-plane. An orthogonal ray graph is a 2-directional orthogonal ray graph if all the horizontal rays extend in the positive x-direction and all the vertical rays extend in the positive y-direction. We first show that the class of orthogonal ray graphs is a proper subset of the class of unit grid intersection graphs. We next provide several characterizations of 2-directional orthogonal ray graphs. Our first characterization is based on forbidden submatrices. A characterization in terms of a vertex ordering follows immediately. Next, we show that 2-directional orthogonal ray graphs are exactly those bipartite graphs whose complements are circular arc graphs. This characterization implies polynomial-time recognition and isomorphism algorithms for 2-directional orthogonal ray graphs. It also leads to a characterization of 2-directional orthogonal ray graphs by a list of forbidden induced subgraphs. We also show a characterization of 2-directional orthogonal ray trees, which implies a linear-time algorithm to recognize such trees. Our results settle an open question of deciding whether a (0,1)-matrix can be permuted to avoid the submatrices .     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.     

Algorithms for clique-independent sets on subclasses of circular-arc graphs

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose polynomial time algorithms for finding the maximum cardinality and weight of a clique-independent set of a -free CA graph. Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is O(n). This represents an improvement over the existing algorithm by Guruswami and Pandu Rangan, whose complexity is O(n²). These algorithms suppose that an HCA model of the graph is given.     

Tutte’s 5-flow conjecture for highly cyclically connected cubic graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω(G) be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph G. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω≥2. We show that if a cubic graph G has no edge cut with fewer than edges that separates two odd cycles of a minimum 2-factor of G, then G has a nowhere-zero 5-flow. This implies that if a cubic graph G is cyclically n-edge connected and , then G has a nowhere-zero 5-flow.     

Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted , of sparse graphs. The maximum average degree of a graph G, denoted mad(G), is the maximum of the average degrees of all subgraphs of G. It is clear that any graph G with maximum degree Δ(G) satisfies . In this paper, we prove the following results: (1) if and Δ(G)≥3, then , and we give an infinite family of examples to show that this result is best possible; (2) if and Δ(G)≥9, then , and we give an infinite family of examples to show that the bound on cannot be increased in general; (3) if G is planar and has girth at least 5, then .     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

The exact domination number of the generalized Petersen graphs

Let G=(V,E) be a graph. A subset S⊆V is a dominating set of G, if every vertex u∈V−S is dominated by some vertex v∈S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2008) 603-610] proved that and conjectured that the upper bound is the exact domination number. In this paper we prove this conjecture.     

Edge intersection graphs of linear 3-uniform hypergraphs

Let be the class of edge intersection graphs of linear 3-uniform hypergraphs. It is known that the problem of recognition of the class is NP-complete. We prove that this problem is polynomially solvable in the class of graphs with minimum vertex degree ≥10. It is also proved that the class is characterized by a finite list of forbidden induced subgraphs in the class of graphs with minimum vertex degree ≥16.     

Acyclic edge colouring of planar graphs without short cycles

Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edgek-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by .In 2001, Alon et al. conjectured that for any graph G it holds that ; here Δ(G) stands for the maximum degree of G.In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then .     

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.