On the nullity and the matching number of unicyclic graphs

Let G be a graph with n vertices and ν(G) be the matching number of G. Let η(G) denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G)=n-2ν(G). Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212-220] proved that the nullity set of all unicyclic graphs with n vertices is {0,1,…,n-4} and characterized the unicyclic graphs with η(G)=n-4. In this paper, we characterize the unicyclic graphs with η(G)=n-5, and we prove that if G is a unicyclic graph, then η(G) equals , or n-2ν(G)+2. We also give a characterization of these three types of graphs. Furthermore, we determine the unicyclic graphs G with η(G)=0, which answers affirmatively an open problem by Tan and Liu.     

Cubicity and Bandwidth

A unit cube in ${\mathbb{R}^k}$ (or a k-cube in short) is defined as the Cartesian product R ₁ × R ₂?× ... × R _k where R _i (for 1??? i??? k) is a closed interval of the form [a _i , a _i + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k-cube representation. From a geometric embedding point of view, a k-cube representation of G?=?(V, E) yields an embedding ${f: V(G) \rightarrow \mathbb{R}^k}$ such that for any two vertices u and v, ||f(u) ? f(v)||_?? ?? 1 if and only if ${(u, v) \in E(G)}$ . We first present a randomized algorithm that constructs the cube representation of any graph on n vertices with maximum degree ?? in O(?? ln n) dimensions. This algorithm is then derandomized to obtain a polynomial time deterministic algorithm that also produces the cube representation of the input graph in the same number of dimensions. The bandwidth ordering of the graph is studied next and it is shown that our algorithm can be improved to produce a cube representation of the input graph G in O(?? ln b) dimensions, where b is the bandwidth of G, given a bandwidth ordering of G. Note that b ?? n and b is much smaller than n for many well-known graph classes. Another upper bound of b + 1 on the cubicity of any graph with bandwidth b is also shown. Together, these results imply that for any graph G with maximum degree ?? and bandwidth b, the cubicity is O(min{b, ?? ln b}). The upper bound of b?+ 1 is used to derive upper bounds for the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs in terms of the maximum degree ??.     

The Four-in-a-Tree Problem in Triangle-Free Graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n ⁴) whether three given vertices of a graph belong to an induced tree. Here, we study four-in- a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the “same structure”, in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.     

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition \(E_1 \cup E_2 \cup \cdots \cup E_k\) of the edge set E(G) such that each \(E_i\) induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.     

On Family of Graphs with Minimum Number of Spanning Trees

We show that there is a well-defined family of connected simple graphs Λ(n, m) on n vertices and m edges such that all graphs in Λ(n, m) have the same number of spanning trees, and if ${G \in \Lambda(n, m)}$ then the number of spanning trees in G is strictly less than the number of spanning trees in any other connected simple graph ${H, H \notin \Lambda(n, m)}$ , on n vertices and m edges.     

On the OBDD size for graphs of bounded tree- and clique-width

We study the size of OBDDs (ordered binary decision diagrams) for representing the adjacency function f_G of a graph G on n vertices. Our results are as follows:

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for graphs of bounded tree-width there is an OBDD of size O(logn) for f_G that uses encodings of size O(logn) for the vertices;

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for graphs of bounded clique-width there is an OBDD of size O(n) for f_G that uses encodings of size O(n) for the vertices;

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for graphs of bounded clique-width such that there is a clique-width expression for G whose associated binary tree is of depth O(logn) there is an OBDD of size O(n) for f_G that uses encodings of size O(logn) for the vertices;

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for cographs, i.e. graphs of clique-width at most 2, there is an OBDD of size O(n) for f_G that uses encodings of size O(logn) for the vertices. This last result complements a recent result by Nunkesser and Woelfel [R. Nunkesser, P. Woelfel, Representation of graphs by OBDDs, in: X. Deng, D. Du (Eds.), Proceedings of ISAAC 2005, in: Lecture Notes in Computer Science, vol. 3827, Springer, 2005, pp. 1132-1142] as it reduces the size of the OBDD by an O(logn) factor using encodings whose size is increased by an O(1) factor.

     

Graphs of nonsolvable groups with four degree-vertices

Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.     

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.     

Construction of a family of graphs with a small induced proper subgraph with minimum degree 3

We investigate the following question proposed by Erd?s: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n-1edges, andδ(G)?3, then G contains an induced proper subgraph H with at least cn vertices andδ(H)?3?Previously we showed that there exists no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K₄. Also a similar construction of a graph with n vertices and αn+β edges is given.     

Domination Defect in Graphs: Guarding With Fewer Guards

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.     

Independence numbers of random subgraphs of distance graphs

We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erd?s–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log₂n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.     

Group distance magic and antimagic graphs

Given a graph G with n vertices and an Abelian group A of order n, an A-distance antimagic labelling of G is a bijection from V (G) to A such that the vertices of G have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of A) of the labels assigned to its neighbours. An A-distance magic labelling of G is a bijection from V (G) to A such that the weights of all vertices of G are equal to the same element of A. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.     

New Ore��s Type Results on Hamiltonicity and Existence of Paths of Given Length in Graphs

The well-known Ore??s theorem (see Ore in Am Math Mon 65:55, 1960), states that a graph G of order n such that d(x)?+?d(y)??? n for every pair {x, y} of non-adjacent vertices of G is Hamiltonian. In this paper, we considerably improve this theorem by proving that in a graph G of order n and of minimum degree ????? 2, if there exist at least n ? ?? vertices x of G so that the number of the vertices y of G non-adjacent to x and satisfying d(x)?+?d(y)??? n ? 1 is at most ?? ? 1, then G is Hamiltonian. We will see that there are graphs which violate the condition of the so called ??Extended Ore??s theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007) as well as the condition of Chvatál??s theorem (see Chvátal in J Combin Theory Ser B 12:163?C168, 1972) and the condition of the so called ??Extended Fan?? theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007), but satisfy the condition of our result, which then, only allows to conclude that such graphs are Hamiltonian. This will show the pertinence of our result. We give also a new result of the same type, ensuring the existence of a path of given length.     

Finding large cycles in Hamiltonian graphs

We show how to find in Hamiltonian graphs a cycle of length n^{Ω(1/loglogn)}=exp(Ω(logn/loglogn)). This is a consequence of a more general result in which we show that if G has a maximum degree d and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in O(n³) time a cycle in G of length k^Ω(1/logd). From this we infer that if G has a cycle of length k, then one can find in O(n³) time a cycle of length k^{Ω(1/(log(n/k)+loglogn))}, which implies the result for Hamiltonian graphs. Our results improve, for some values of k and d, a recent result of Gabow (2004) [11] showing that if G has a cycle of length k, then one can find in polynomial time a cycle in G of length . We finally show that if G has fixed Euler genus g and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in polynomial time a cycle in G of length f(g)k^Ω(1), running in time O(n²) for planar graphs.     

Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

The open neighborhood N(v) of a vertex v in a graph G is the set of vertices adjacent to v in G. A graph is twin-free (or open identifiable) if every two distinct vertices have distinct open neighborhoods. A separating open code in G is a set C of vertices such that \({N(u) \cap C \neq N(v) \cap C}\) for all distinct vertices u and v in G. An open dominating set, or total dominating set, in G is a set C of vertices such that \({N(u) \cap C \ne N(v) \cap C}\) for all vertices v in G. An identifying open code of G is a set C that is both a separating open code and an open dominating set. A graph has an identifying open code if and only if it is twin-free. If G is twin-free, we denote by \({\gamma^{\rm IOC}(G)}\) the minimum cardinality of an identifying open code in G. A hypergraph H is identifiable if every two edges in H are distinct. A distinguishing-transversal T in an identifiable hypergraph H is a subset T of vertices in H that has a nonempty intersection with every edge of H (that is, T is a transversal in H) such that T distinguishes the edges, that is, \({e \cap T \neq f \cap T}\) for every two distinct edges e and f in H. The distinguishing-transversal number \({\tau_D(H)}\) of H is the minimum size of a distinguishing-transversal in H. We show that if H is a 3-uniform identifiable hypergraph of order n and size m with maximum degree at most 3, then \({20\tau_D(H) \leq 12n + 3m}\) . Using this result, we then show that if G is a twin-free cubic graph on n vertices, then \({\gamma^{\rm IOC}(G) \leq 3n/4}\) . This bound is achieved, for example, by the hypercube.     

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 ordering of unicyclic graphs with the smallest algebraic connectivity

Fielder [M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305] has turned out that G is connected if and only if its algebraic connectivity a(G)>0. In 1998, Fallat and Kirkland [S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74] posed a conjecture: if G is a connected graph on n vertices with girth g≥3, then a(G)≥a(C_n,g) and that equality holds if and only if G is isomorphic to C_n,g. In 2007, Guo [J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with vertices.     

An upper bound for Cubicity in terms of Boxicity

An axis-parallel b-dimensional box is a Cartesian product R₁×R₂×?×R_b where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,b_i] on the real line. The boxicity of any graph G, is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R₁×R₂×?×R_b, where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,a_i+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by ). In this paper we prove that , where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below:

1.

Planar graphs have cubicity at most 3⌈log₂n⌉.

2.

Outer planar graphs have cubicity at most 2⌈log₂n⌉.

3.

Any graph of treewidth tw has cubicity at most (tw+2)⌈log₂n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log₂n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log₂n⌉, where ω is the clique number.

The above upper bounds are tight, but for small constant factors.