Long cycles in bipartite graphs

Let G be a 2-connected bipartite graph with bipartition (A, B), where |A| ≥ |B|. It is shown that if each vertex of A has degree at least k, and each vertex of B has degree at least l, then G contains a cycle of length at least 2min(|B|, k + l ? 1, 2k ? 2). Then this result is used to determine the minimum number of edges required in a bipartite graph to ensure a cycle of length at least 2m, for any integer m ≥ 2.     

On bipartite graphs of diameter 3 and defect 2

We consider bipartite graphs of degree Δ≥2, diameter D=3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (Δ, 3, ?2) ‐graphs. We prove the uniqueness of the known bipartite (3, 3, ?2) ‐graph and bipartite (4, 3, ?2)‐graph. We also prove several necessary conditions for the existence of bipartite (Δ, 3, ?2) ‐graphs. The most general of these conditions is that either Δ or Δ?2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when Δ=6 and Δ=9, we prove the non‐existence of the corresponding bipartite (Δ, 3, ?2)‐graphs, thus establishing that there are no bipartite (Δ, 3, ?2)‐graphs, for 5≤Δ≤10. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 271–288, 2009     

On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

Menger's Theorem for Graphs Containing no Infinite Paths

Menger's theorem can be stated as follows: Let G = (V, E) be a finite graph, and let A and B be subsets of V. Then there exists a family F of vertex-disjoint paths from A to B and a subset S of V which separates A and B, such that S consists of a choice of precisely one vertex from each path in F.Erdös conjectured that in this form the theorem can be extended to infinite graphs. We prove this to be true for graphs containing no infinite paths, by showing that in this case the problem can be reduced to the case of bipartite graphs.     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].     

2K2 vertex-set partition into nonempty parts

A graph is 2K₂-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K₂-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We show that for C₄-free graphs, circular-arc graphs, spiders, P₄-sparse graphs, and bipartite graphs the 2K₂-partition problem can be solved in polynomial time.     

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph).Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)?1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.     

Some connected ramsey numbers

A graph G is co-connected if both G and its complement ? are connected and nontrivial. For two graphs A and B, the connected Ramsey number r_c(A, B) is the smallest integer n such that there exists a co-connected graph of order n, and if G is a co-connected graph on at least n vertices, then A ? G or B ? ?. If neither A or B contains a bridge, then it is known that r_c(A, B) = r(A, B), where r(A, B) denotes the usual Ramsey number of A and B. In this paper r_c(A, B) is calculated for some pairs (A, B) when r(A, B) is known and at least one of the graphs A or B has a bridge. In particular, r_c(A, B) is calculated for A a path and B either a cycle, star, or complete graph, and for A a star and B a complete graph.     

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.      

On compact and efficient routing in certain graph classes

In this paper we refine the notion of tree-decomposition by introducing acyclic (R,D)-clustering, where clusters are subsets of vertices of a graph and R and D are the maximum radius and the maximum diameter of these subsets. We design a routing scheme for graphs admitting induced acyclic (R,D)-clustering where the induced radius and the induced diameter of each cluster are at most 2. We show that, by constructing a family of special spanning trees, one can achieve a routing scheme of deviation Δ?2R with labels of size bits per vertex and O(1) routing protocol for these graphs. We investigate also some special graph classes admitting induced acyclic (R,D)-clustering with induced radius and diameter less than or equal to 2, namely, chordal bipartite, homogeneously orderable, and interval graphs. We achieve the deviation Δ=1 for interval graphs and Δ=2 for chordal bipartite and homogeneously orderable graphs.     

Towards a spectral theory of graphs based on the signless Laplacian, II

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.     

Linear choosability of graphs

A proper vertex coloring of a non-oriented graph G is linear if the graph induced by the vertices of any two color classes is a forest of paths. A graph G is linearly L-list colorable if for a given list assignment L={L(v):v∈V(G)}, there exists a linear coloring c of G such that c(v)∈L(v) for all v∈V(G). If G is linearly L-list colorable for any list assignment with |L(v)|?k for all v∈V(G), then G is said to be linearly k-choosable. In this paper, we investigate the linear choosability for some families of graphs: graphs with small maximum degree, with given maximum average degree, outerplanar and planar graphs. Moreover, we prove that deciding whether a bipartite subcubic planar graph is linearly 3-colorable is an NP-complete problem.     

On direct product cancellation of graphs

The direct product of graphs obeys a limited cancellation property. Lovász proved that if C has an odd cycle then A×C≅B×C if and only if A≅B, but cancellation can fail if C is bipartite. This note investigates the ways cancellation can fail. Given a graph A and a bipartite graph C, we classify the graphs B for which A×C≅B×C. Further, we give exact conditions on A that guarantee A×C≅B×C implies A≅B. Combined with Lovász’s result, this completely characterizes the situations in which cancellation holds or fails.     

Graph decomposition with applications to subdivisions and path systems modulo k

The existence of a function α(k) (where k is a natural number) is established such that the vertex set of any graph G of minimum degree at least α(k) has a decomposition A ∪ B ∪ C such that G(A) has minimum degree at least k, each vertex of A is joined to at least k vertices of B, and no two vertices of B are separated by fewer than k vertices in G(G ∪ C). This is applied to prove the existence of subdivisions of complete bipartite graphs (complete graphs) with prescribed path lengths modulo k in graphs of sufficiently high minimum degree (chromatic number) and path systems with prescribed ends and prescribed lengths modulo k in graphs of sufficiently high connectivity.     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

On the sum of the Laplacian eigenvalues of a tree

Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10].     

Non-existence of bipartite graphs of diameter at least 4 and defect 2

The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Δ and diameter D. Bipartite graphs of maximum degree Δ, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Δ such that Δ−1 is a prime power.     

Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

An interval coloring of a graph is a proper edge coloring such that the set of used colors at every vertex is an interval of integers. Generally, it is an NP‐hard problem to decide whether a graph has an interval coloring or not. A bipartite graph G = (A,B;E) is (α, β)‐biregular if each vertex in A has degree α and each vertex in B has degree β. In this paper we prove that if the (3,4)‐biregular graph G has a cubic subgraph covering the set B then G has an interval coloring. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 122–128, 2004