A NOTE ON GRAPHS WHOSE DESTRUCTIONS YIELD COMPLETE SUBGRAPHS

Abstract

A connected graph G of order p =|V| and sise q =| E | is said to be (a_i, b_i)-destructible (with respect to E_i and V_i say) if a_i,b_i are integral factors of p and an a_i-set of edges E_i exists whose removal from G results in exactly b_i components isomorphic to K_i i.e. whose removal from G isolates the vertices in a b_i-set V_i. The operation of removing E_i and V_i from G results in either Ø or a subgraph H of G and is called an (a_i , b_i)-destruction of G. In this paper we show that the only graphs whose every (a_i,b_i)- destruction results in a complete subgraph are K (1,2) and K₄—e, where e ε K₄.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non-bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

On Hypohamiltonian and Almost Hypohamiltonian Graphs

A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that is non‐hamiltonian, and for any vertex the graph is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every . During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.     

Further results on monotonic graph invariants and bipartiteness number

Abstract

The bipartiteness of a graph is the minimum number of vertices whose deletion from G results in a bipartite graph. If a graph invariant decreases or increases with addition of edges of its complement, then it is called a monotonic graph invariant. In this article, we determine the extremal values of some famous monotonic graph invariants, and characterize the corresponding extremal graphs in the class of all connected graphs with a given vertex bipartiteness.     

A note on strong embeddings of maximal planar graphs on non-orientable sufraces

Abstract. In this paper, it is shown that for every maximal planar graph     

Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let C_n denote the cycle of order n and the graph obtained from joining two cycles C₆ by a path P_n-12 with its two leaves. Let B_n denote the class of all bipartite bicyclic graphs but not the graph R_a,b, which is obtained from joining two cycles C_a and C_b (a,b≥10 and ) by an edge. In [I. Gutman, D. Vidovi?, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41(2001) 1002-1005], Gutman and Vidovi? conjectured that the bicyclic graph with maximal energy is , for n=14 and n≥16. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427(2007) 87-98], Li and Zhang showed that the conjecture is true for graphs in the class B_n. However, they could not determine which of the two graphs R_a,b and has the maximal value of energy. In [B. Furtula, S. Radenkovi?, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008) 431-433], numerical computations up to a+b=50 were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of is larger than that of R_a,b, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.     

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n–1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n–1). This generalizes, to weighted graphs, a classical result of Erds and Gallai [4].     

Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG ⁺G, whereG ⁺ is the bipartite multigraph whose adjacency matrix,B ⁺, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.     

Rubber bands,convex embeddings and graph connectivity

We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and physical properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ^k–1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium.As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allkn, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.     

Fractional perfect b-matching polytopes I: General theory

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v   is a prescribed nonnegative number _bv$b_v$. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.     

The gallai-younger conjecture for planar graphs

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG–X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.     

On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

If G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

The bondage numbers of graphs with small crossing numbers

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. Kang and Yuan proved b(G)?8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results to connected graphs with small crossing numbers.     

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

W v cycles in plane graphs

A cycle in a plane graphG is called aW _v cycle if it has a connected (or empty) intersection with each face of the graph. We show that if the minimum degree (G)3 thenG has aW _v cycle and the lengthw(G) of a longestW _v cycle is bounded by the number,f(G), of faces ofG. The classW of graphsG withw(G)=f(G) is completely characterized by an characterized by an inductive construction from two graphs, namelyK ₄ and a face merging of two copies ofK ₄ on one hand, and in terms involving Halin graphs and face merging on the other hand. Longest cycles in members ofW are investigated. The shortness coefficient ofW is proved to be between one-half and three-quarters inclusively.     

Disproof of a conjecture in graph reconstruction theory

In his thesis [3] B. D. Thatte conjectured that ifG=G ₁,G ₂,...G _n is a sequence of finitely many simple connected graphs (isomorphic graphs may occur in the sequence) with the same number of vertices and edges then their shuffled edge deck uniquely determines the graph sequence (up to a permutation). In this paper we prove that there are such sequences of graphs with the same shuffled edge deck.This research was partially supported by Hungarian National Foundation of Scientific Research Grant no. 1812     

Small cycle double covers of 4-connected planar graphs

Acycle double cover of a graph,G, is a collection of cycles,C, such that every edge ofG lies in precisely two cycles ofC. TheSmall Cycle Double Cover Conjecture, proposed by J. A. Bondy, asserts that every simple bridgeless graph onn vertices has a cycle double cover with at mostn–1 cycles, and is a strengthening of the well-knownCycle Double Cover Conjecture. In this paper, we prove Bondy's conjecture for 4-connected planar graphs.     

Graph congruences and what they connote

Abstract

The algebraic notion of a “congruence” seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph G = (V, E) being a pair (～, ) of which ～ is an equivalence relation on V and is a set of unordered pairs of vertices of G with a special relationship to ～ and E. Kernels and quotient structures are used in this theory to develop homomorphism and isomorphism theorems which remind one of similar results in an algebraic context. We show that this theory can be applied to deliver structural decompositions of graphs into “factor” graphs having very special properties, such as the result that each graph, except one, is a subdirect product of graphs with universal vertices. In a final section, we discuss corresponding concepts and briefly describe a corresponding theory for graphs which have a loop at every vertex and which we call loopy graphs. They are in a sense more “algebraic” than simple graphs, with their meet-semilattices of all congruences becoming complete algebraic lattices.