Degree sequences and edge connectivity

For each positive integer k, we give a finite list C(k) of Bondy–Chvátal type conditions on a nondecreasing sequence \(d=(d_1,\dots ,d_n)\) of nonnegative integers such that every graph on n vertices with degree sequence at least d is k-edge-connected. These conditions are best possible in the sense that whenever one of them fails for d then there is a graph on n vertices with degree sequence at least d which is not k-edge-connected. We prove that C(k) is and must be large by showing that it contains p(k) many logically irredundant conditions, where p(k) is the number of partitions of k. Since, in the corresponding classic result on vertex connectivity, one needs just one such condition, this is one of the rare statements where the edge connectivity version is much more difficult than the vertex connectivity version. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. We prove that any sublist equivalent to C(k) has length at least p(k), where p(k) is the number of partitions of k, which is in contrast to the corresponding classic result on vertex connectivity where one needs just one such condition. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. Finally, we informally describe a simple and fast procedure which generates the list C(k). Specialized to \(k=3\), this verifies a conjecture of Bauer, Hakimi, Kahl, and Schmeichel, and for \(k=2\) we obtain an alternative proof for their result on bridgeless connected graphs. The explicit list for \(k=4\) is given, too.     

Hamiltonian Type Properties in Claw-Free $$P_5$$-Free Graphs

A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each integer r in the set \(\{ m, m + 1, \ldots , n \}\). This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree et al. in (Graphs Combin 20:291–310, 2004). The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. Broersma and Veldman showed in (Contemporary methods in graph theory, BI-Wiss.-Verlag, Mannheim, Wien, Zürich, pp 181–194, 1990) that any 2-connected claw-free \(P_5\)-free graph must be hamiltonian. In fact, every non-hamiltonian cycle in such a graph is either extendable or very dense. We show that any 2-connected claw-free \(P_5\)-free graph is (k, 3k)-pancyclic for each integer \(k \ge 2\). We also show that such a graph is (1, 5)-pancyclic. Examples are provided which show that these results are best possible. Each example we provide represents an infinite family of graphs.     

The Number of Labeled Outerplanar <Emphasis Type="Italic">k</Emphasis>-Cyclic Graphs

A k-cyclic graph is a graph with cyclomatic number k. An explicit formula for the number of labeled connected outerplanar k-cyclic graphs with a given number of vertices is obtained. In addition, such graphs with fixed cyclomatic number k and a large number of vertices are asymptotically enumerated. As a consequence, it is found that, for fixed k, almost all labeled connected outerplanar k-cyclic graphs with a large number of vertices are cacti.     

Edge-transitive products

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a strong product is edge-transitive if and only if all factors are complete graphs. In addition, a connected, infinite non-trivial Cartesian product graph G is edge-transitive if and only if it is vertex-transitive and if G is a finite weak Cartesian power of a connected, edge- and vertex-transitive graph H, or if G is the weak Cartesian power of a connected, bipartite, edge-transitive graph H that is not vertex-transitive.     

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.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

Chromatic number and subtrees of graphs

Let G and H be two graphs. We say that G induces H if G has an induced subgraph isomorphic to H: A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T. there exists a function f _T ; called binding function, depending only on T with the property that every graph G with chromatic number f _T (ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza confirmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer k and every tree T of radius r, every graph G with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14 ^r-1(r - 1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r, every graph with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6 ^r?2) times.     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

<Emphasis Type="Italic">k</Emphasis>-Trails: recognition,complexity, and approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail, and every graph containing a k-trail is a \((k+1)\)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a \((2k-1)\)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnár, Newman, and Seb?.     

Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ <Emphasis Type="Italic">α</Emphasis> &lt; 0

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randi? index \(R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} \), where d_G(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randi? index for ?1 ≤ α < 0, respectively.     

A cycle cover of a 2-edge-connected graph embedded with large face-width on an orientable surface

In 1985, Alon and Tarsi conjectured that the length of a shortest cycle cover of a bridgeless graph H is at most 7/5 |E(H|). The conjecture is still open. Let G be a 2-edge-connected graph embedded with face-width k on the non-spherical orientable surface S_g. We give an upper bound on the length of a cycle cover of G. In particular, if g = 1 and k ≥ 48, or g = 2 and k ≥ 427, or g ≥ 3 and k ≥ 288(4^g - 1), then the upper bound is 7/5 |E(G|), which means that Alon and Tarsi’s conjecture holds for such a graph.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Edge-colouring of graphs and hereditary graph properties

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G, a hereditary graph property P and l ? 1 we define \(X{'_{P,l}}\)(G) to be the minimum number of colours needed to properly colour the edges of G, such that any subgraph of G induced by edges coloured by (at most) l colours is in P. We present a necessary and sufficient condition for the existence of \(X{'_{P,l}}\)(G). We focus on edge-colourings of graphs with respect to the hereditary properties O_k and S_k, where O_k contains all graphs whose components have order at most k+1, and S_k contains all graphs of maximum degree at most k. We determine the value of \(X{'_{{S_k},l}}(G)\) for any graph G, k ? 1, l ? 1, and we present a number of results on \(X{'_{{O_k},l}}(G)\).     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

The graph Kre(4) does not exist

Suppose that a strongly regular graph Γ with parameters (v, k, λ, μ) has eigenvalues k, r, and s. If the graphs Γ and \(\bar \Gamma \) are connected, then the following inequalities, known as Krein’s conditions, hold: (i) (r + 1)(k + r + 2rs) ≤ (k + r)(s + 1)² and (ii) (s + 1)(k + s + 2rs) ≤ (k + s)(r + 1)². We say that Γ is a Krein graph if one of Krein’s conditions (i) and (ii) is an equality for this graph. A triangle-free Krein graph has parameters ((r ² + 3r)², r ³ + 3r ² + r, 0, r ² + r). We denote such a graph by Kre(r). It is known that, in the cases r = 1 and r = 2, the graphs Kre(r) exist and are unique; these are the Clebsch and Higman–Sims graphs, respectively. The latter was constructed in 1968 together with the Higman–Sims sporadic simple group. A.L. Gavrilyuk and A.A. Makhnev have proved that the graph Kre(3) does not exist. In this paper, it is proved that the graph Kre(4) (a strongly regular graph with parameters (784, 116, 0, 20)) does not exist either.     

On the minimum-cost $$\lambda $$-edge-connected k-subgraph problem

In this paper, we propose several integer programming (IP) formulations to exactly solve the minimum-cost \(\lambda \)-edge-connected k-subgraph problem, or the \((k,\lambda )\)-subgraph problem, based on its graph properties. Special cases of this problem include the well-known k-minimum spanning tree problem (if \(\lambda =1\)), \(\lambda \)-edge-connected spanning subgraph problem (if \(k=|V|\)) and k-clique problem (if \(\lambda = k-1\) and there are exact k vertices in the subgraph). As a generalization of k-minimum spanning tree and a case of the \((k,\lambda )\)-subgraph problem, the (k, 2)-subgraph problem is studied, and some special graph properties are proved to find stronger and more compact IP formulations. Additionally, we study the valid inequalities for these IP formulations. Numerical experiments are performed to compare proposed IP formulations and inequalities.     

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.     

Note on a conjecture for the sum of signless Laplacian eigenvalues

For a simple graph G on n vertices and an integer k with 1 ? k ? n, denote by \(\mathcal{S}^+_k\) (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that \(\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})\) (G) ? e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n ? 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n ? 2, and for some new classes of graphs.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.