The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

Degree-associated reconstruction number of graphs

A card of a graph G is a subgraph formed by deleting one vertex. The Reconstruction Conjecture states that each graph with at least three vertices is determined by its multiset of cards. A dacard specifies the degree of the deleted vertex along with the card. The degree-associated reconstruction number drn(G) is the minimum number of dacards that determine G. We show that drn(G)=2 for almost all graphs and determine when drn(G)=1. For k-regular n-vertex graphs, drn(G)≤min{k+2,n−k+1}. For vertex-transitive graphs (not complete or edgeless), we show that drn(G)≥3, give a sufficient condition for equality, and construct examples with large drn. Our most difficult result is that drn(G)=2 for all caterpillars except stars and one 6-vertex example. We conjecture that drn(G)≤2 for all but finitely many trees.     

On the extremal Merrifield-Simmons index and Hosoya index of quasi-tree graphs

It is well known that the two graph invariants, “the Merrifield-Simmons index” and “the Hosoya index” are important in structural chemistry. A graph G is called a quasi-tree graph, if there exists u₀ in V(G) such that G−u₀ is a tree. In this paper, at first we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield-Simmons indices. Then we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those n-vertex quasi-tree graphs with k pendent vertices having the smallest Hosoya index.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

A Note on Turán Numbers for Even Wheels

The Turán number ex(n, G) is the maximum number of edges in any n-vertex graph that does not contain a subgraph isomorphic to G. We consider a very special case of the Simonovits’s theorem (Simonovits in: Theory of graphs, Academic Press, New York, 1968) which determine an asymptotic result for Turán numbers for graphs with some properties. In the paper we present a more precise result for even wheels. We provide the exact value for Turán number ex(n, W _2k ) for n ≥ 6k ? 10 and k ≥ 3. In addition, we show that ${ex(n,W_6)= \lfloor\frac{n^2}{3}\rfloor}$ for all n ≥ 6. These numbers can be useful to calculate some Ramsey numbers.     

On the signless Laplacian index of cacti with a given number of pendant vertices

A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest signless Laplacian index among all the cacti with n vertices and k pendant vertices. As a consequence, we determine the graph with the largest signless Laplacian index among all the cacti with n vertices; we also characterize the n-vertex cacti with a perfect matching having the largest signless Laplacian index.     

Tree-decompositions with bags of small diameter

This paper deals with the length of a Robertson-Seymour's tree-decomposition. The tree-length of a graph is the largest distance between two vertices of a bag of a tree-decomposition, minimized over all tree-decompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded tree-length graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, AT-free graphs, etc.). For instance, we show that the tree-length of any outerplanar graph is ⌈k/3⌉, where k is the chordality of the graph, and we compute the tree-length of meshes.More fundamentally we show that any algorithm computing a tree-decomposition approximating the tree-width (or the tree-length) of an n-vertex graph by a factor α or less does not give an α-approximation of the tree-length (resp. the tree-width) unless if α=Ω(n^1/5). We complete these results presenting several polynomial time constant approximate algorithms for the tree-length.The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with near-optimal route length, and by the construction of sparse additive spanners.     

The basis number of a graph

MacLane proved that a graph is planar if and only if it has a 2-fold basis for its cycle space. We define the basis number of a graph G to be the least integer k such that G has a k-fold basis for its cycle space. We investigate the basis number of the complete graphs, complete bipartite graphs, and the n-cube.     

Queue layouts of iterated line directed graphs

In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph L^k(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of L^k(G), it is shown that for any fixed directed graph G, L^k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L^k(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd.     

On the nullity of graphs with pendent vertices

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60-67] characterized the extremal graphs attaining the upper bound n-2 and the second upper bound n-3. In this paper, as the continuance of it, we determine the extremal graphs with pendent vertices achieving the third upper bound n-4 and fourth upper bound n-5. We then proceed recursively to construct all graphs with pendent vertices which satisfy η(G)>0. Our results provide a unified approach to determine n-vertex unicyclic (respectively, bicyclic and tricyclic) graphs which achieve the maximal and second maximal nullity and characterize n-vertex extremal trees attaining the second and third maximal nullity. As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.     

Results on the Grundy chromatic number of graphs

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i<j, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by Γ(G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining Γ(G)?k, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus-Gaddum-type inequalities concerning Γ(G) and Γ(G^c), for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with χ(G) colors. Equivalently G is well colored if Γ(G)=χ(G). We prove that the recognition problem of well-colored graphs is a coNP-complete problem.     

Characterizations of 2-variegated graphs and of 3-variegated graphs

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. We prove that a graph G on 2n vertices is 2-variegated if and only if there exists a set S of n independent edges in G such that no cycle in G contains an odd number of edges from S. We also characterize 3-variegated graphs.     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.     

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph has been studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is not smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.