4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to K_1,3. The kth power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are , and T_r, r?2, where T_r is the graph with 4r vertices u_i,v_i,x_i,y_i, 1?i?r, such that for each i with 1?i?r, {u_i,v_i,x_i,y_i} is a clique of T_r and . We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K₄×K₂, and .     

Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Horňák, Kalinowski, Meszka and Wo?niak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?     

Recursive construction for 3-regular expanders

We present an algorithm which inn ³ (logn)³ time constructs a 3-regular expander graph onn vertices. In each step we substitute a pair of edges of the graph by a new pair of edges so that the total number of cycles of lengths=clogn decreases (for some fixed absolute constantc). When we reach a local minimum in the number of cycles of lengths the graph is an expander.     

Pairs of edge disjoint Hamiltonian circuits in 5-connected planar graphs

Summary A variety of examples of 4-connected 4-regular graphs with no pair of disjoint Hamiltonian circuits were constructed in response to Nash-Williams conjecture that every 4-connected 4-regular graph is Hamiltonian and also admits a pair of edge-disjoint Hamiltonian circuits. Nash-Williams's problem is especially interesting for planar graphs since 4-connected planar graphs are Hamiltonian. Examples of 4-connected 4-regular planar graphs in which every pair of Hamiltonian circuits have edges in common are included in the above mentioned examples.B. Grünbaum asked whether 5-connected planar graphs always admit a pair of disjoint Hamiltonian circuits. In this paper we introduce a technique that enables us to construct infinitely many examples of 5-connected planar graphs, 5-regular and non regular, in which every pair of Hamiltonian circuits have edges in common.     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

On the skew energy of orientations of hypercubes

Let D be a digraph of order n and λ₁,λ₂,…,λ_n denote all the eigenvalues of the skew-adjacency matrix of D. The skew energy E_S(D) of D is defined as . In this paper, it is proved that for any positive integer k≥3, there exists a k-regular graph of order n having an orientation D with . This work positively answers a problem proposed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a digraph, Linear Algebra Appl. 432 (2010) 1825-1835]. In addition, a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph.     

Extensions on 2-edge connected 3-regular up-embeddable graphs

1.IntroductionLetHbea3-regulargraph.Forel,e2,e3EE(H)(el,eZande3areallowedtobethesame),weaddthreenewvenicesal)wZandw3inel,eZande3respectively.ChoosingueV(H),andthenjoiningutofi(i=1,2,3),weatlastobtaina3-regulargraphG.Or,inotherwords,wesaythatGisobtainedfromHbyanM-extension.DenoteG=M(u)(H)(seeFig.1).LetHbea3-regulargraph.Takingel,eZEE(H)(elandeZareallowedtobethesame),weputtwodistinctvenicestvian0dwZinelandeZrespectively,andaddtwodistinctvenicesu,v4V(H),thenjoinutoal,joinvtowZandjoin…     

Extremal Problems For Transversals In Graphs With Bounded Degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property , we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property . In this paper we study this problem for the following properties : “acyclic”, “H-free”, and “having connected components of order at most r”. We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+⌞d/r⌟, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner’s Lemma. We also establish some limitations on the power of this topological method. We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of Bollobás, Erdős and Szemerédi. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph. Finally, we pose several open questions. * Research supported by the joint Berlin/Zurichgrad uate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich. † Research partially supported by Hungarian National Research Fund grants T-037846, T-046234 and AT-048826.     

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range (if true, this is tight, e.g. by the Alon–Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue . The proof uses the following lemma (Lemma 3.3): Let A be a real symmetric matrix with zeros on the diagonal. Let d be such that the l₁ norm of each row in A is at most d. Suppose that for every x,y ∈{0,1}ⁿ with ‹x,y›=0. Then the spectral radius of A is O(α(log(d/α)+1)). An interesting consequence of this lemma is a converse to the Expander Mixing Lemma. * This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.     

A lower bound for the spectral radius of a digraph

We show that the spectral radius ρ(D) of a digraph D with n vertices and c₂ closed walks of length 2 satisfies Moreover, equality occurs if and only if D is the symmetric digraph associated to a -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.     

Note on the 3-graph counting lemma

Szemerédi's regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a k-partite graph with k-partition V₁∪?∪V_k, |V₁|=?=|V_k|=n, where all induced bipartite graphs G[V_i,V_j] are (d,ε)-regular, then the number of k-cliques K_k in G is . Frankl and Rödl extended Szemerédi's regularity lemma to 3-graphs and Nagle and Rödl established an accompanying 3-graph counting lemma analogous to the graph counting lemma above. In this paper, we provide a new proof of the 3-graph counting lemma.     

A cycle-space invariant of the <2-distance-graph in the plane

By exhibiting a certain invariant, we prove that the cycle space of the distance<2 graph in the plane is not generated by the triangles inscribed in unit circles. This solves a problem of Lovász in the negative.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

A Two-Variable Interlace Polynomial

We introduce a new graph polynomial in two variables. This interlace polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix.The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs.We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the vertex-nullity interlace polynomial, is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte–Martin polynomial on isotropic systems previously considered by Bouchet. Another, the vertex-rank interlace polynomial, is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial. Supported by NSF grant DMS-9971788.     

Bipartite subgraphs

It is shown that there exists a positivec so that for any large integerm, any graph with 2m ²edges contains a bipartite subgraph with at least edges. This is tight up to the constantc and settles a problem of Erdös. It is also proved that any triangle-free graph withe>1 edges contains a bipartite subgraph with at least e/2+c e ^4/5 edges for some absolute positive constantc. This is tight up to the constantc.Research supported in part by a USA Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

The cover pebbling number of graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.     

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let denote the number of convex cycles of a simple graph G of order n, size m, and girth . It is proved that and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.