Quotients of connected regular graphs of even degree

This paper introduces a method of listing all nonequivalent quotients of any connected regular graph of even degree with a given 2-factorization. The method is based on the characterization of connected 2d-regular graphs as Schreier coset graphs given by Gross (J. Combin. Theory Ser. B22 (1977), 227–232). Various representations of a given graph with a fixed 2-factorization are also investigated. The work is related to graph imbedding theory, particularly to voltage and current graphs.     

Symmetry Groups of Coloured Graphs

A perfect colouring Φ of a simple undirected connected graph G is an edge colouring such that each vertex is incident with exactly one edge of each colour. This paper concerns the problem of representing groups by graphs with perfect colourings. We define groups of graph automorphisms, which preserve the structure of the colouring, and characterize these groups up to isomorphism. Our considerations are based on the fact that every perfectly coloured graph is isomorphic to a Schreier coset graph on a group generated by involutions.     

Asymmetric Combinatorially-Regular Maps

It is shown that for every g≥3, there exists a combinatorially regular map M of type (3, 7) on a closed orientable surface of genus g, such that M has trivial symmetry group. Such maps are constructed from Schreier coset graphs corresponding to permutation representations of the (2, 3, 7) triangle group. 1991 Mathematics Subject Classification: 57M15.     

On the coset graph construction of distance-regular graphs

We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regular codes over finite fields.     

A sufficient condition for a regular graph to be class 1

The core GΔ of a simple graph G is the subgraph induced by the vertices of maximum degree. It is well known that the Petersen graph is not 1-factorizable and has property that the core of the graph obtained from it by removing one vertex has maximum degree 2. In this paper, we prove the following result. Let G be a regular graph of even order with d(G) ≥ 3. Suppose that G contains a vertex ν such that the core of G\ν has maximum degree 2. If G is not the Petersen graph, then G is 1-factorizable. © 1993 John Wiley & Sons, Inc.     

A strongly regularN-full graph of small order

For every positive integern we show the construction of a strongly regular graph of order at most 2 ⁿ⁺² which contains every graph of ordern as a subgraph. The estimation concerning the construction is best possible.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.     

A minimum degree condition forcing complete graph immersion

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P _uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P _uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K _t . For dense graphs one can say even more. If the graph has order n and has 2cn ² edges, then there is a strong immersion of the complete graph on at least c ² n vertices in G in which each path P _uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd ^3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t?1 contains an immersion of K _t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.     

On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k ? 1) + ... + k(k ? 1)^d?1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k ² + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).     

Impact of memory size on graph exploration capability

A mobile agent (robot), modeled as a finite automaton, has to visit all nodes of a regular graph. How does the memory size of the agent (the number of states of the automaton) influence its exploration capability? In particular, does every increase of the memory size enable an agent to explore more graphs? We give a partial answer to this problem by showing that a strict gain of the exploration power can be obtained by a polynomial increase of the number of states. We also show that, for automata with few states, the increase of memory by even one state results in the capability of exploring more graphs.     

Cayley图的泛因子性质

一个κ-正则图若满足对任意正整数s,1≤s≤κ,均存在一个s-因子或一个2[s/2]因子,则称其有泛因子或偶泛因子性质.本文证明了每个奇度Cayley图是泛因子的,每个偶度Carley图是偶泛因子的.同时证明了二面体群上的每个Cayley图均是泛因子的.     

The square of every two-connected graph is Hamiltonian

A graph G is a line-critical block if κ(G) = 2 and if for any line e of G the graph G ? e has κ(G ? e) = 1.If G is a line-critical block, then G is either a DT-block (i.e., G is a two-connected graph in which every line is incident to a point of degree two), or G contains a specific two-connected subgraph which is a DT-block (Theorem 1). Using this result and results of the preceding paper on DT-graphs, a simple proof of the conjecture that the square of every two-connected graph is Hamiltonian is given.     

6‐regular 4‐critical graph

P. Erd?s conjectured in [2] that r‐regular 4‐critical graphs exist for every r ≥ 3 and noted that no such graphs are known for r ≥ 6. This article contains the first example of a 6‐regular 4‐critical graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 286–291, 2002     

Covering the vertex set of a graph with subgraphs of smaller degree

The problem of covering the vertex set of a graph with subsets spanning subgraphs of smaller degree is studied. The result of this study is applied to give a new bound on the chromatic number of a graph in terms of the maximum vertex-degree of the graph and the maximum number of vertices in a clique of the graph. By using this bound, it is shown that if d is at least 7 and e is at least 4, then there is no regular graph of valency d, chromatic number d, whose smallest circuit has at least e edges; this settles a conjecture of Branko Grünbaum.     

Even edge colorings of a graph

It is shown that the minimum number of colors needed to paint the edges of a graph G so that in every cycle of G there is a nonzero even number of edges of at least one color is ?log_2χ(G)?.     

Every finite graph is a full subgraph of a rigid graph

We prove that every finite undirected graph is a full subgraph of a rigid graph. Our construction proceeds on taking a family of “mutually rigid” graphs and attaching them to the vertices of a given graph in a one-to-one manner; then the vertices are fixed on their place. Actually, the new graph is “strongly rigid”, which enables us to show that the category of all graphs containing a given finite graph as a full subgraph is binding.     

Decomposing a graph into expanding subgraphs

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following:

• A classical result of Lipton, Rose and Tarjan from 1979 states that if is a hereditary family of graphs and every graph in has a vertex separator of size , then every graph in has O(n) edges. We construct a hereditary family of graphs with vertex separators of size such that not all graphs in the family have O(n) edges.
• Trevisan and Arora‐Barak‐Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges.
• Sudakov and the second author have recently shown that every graph with average degree d contains an n‐vertex subgraph with average degree at least and vertex expansion . We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1).

The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super‐linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.     

On edge connectivity and parity factor

By Petersen's Theorem, a bridgeless cubic graph has a 2-factor. Fleischner(Discrete Math.,101, 33–37(1992)) has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let me 0 be even and mo 0 be odd. In this paper, we prove that every m e-edge-connected graph with minimum degree at least me + 1 contains an even factor with minimum degree at least m e and every(m o + 1)-edge-connected graph contains an odd factor with minimum degree at least m o, which further extends Fleischner's result. Moreover, we show that our results are best possible.     

On the nullity of a graph with cut-points

Let G be a simple graph of order n and A(G) be its adjacency matrix. The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of A(G). Denote by Ck and Lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in Ck, respectively. In 1998, Sciriha [I. Sciriha, On singular line graphs of trees, Congr. Numer. 135 (1998) 73-91] show that the order of every tree whose line graph is singular is even. Then Gutman and Sciriha [I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35-45] show that the nullity set of L0 is {0,1}. In this paper, we investigate the nullity of graphs with cut-points and deduce some concise formulas. Then we generalize Scirihas' result, showing that the order of every graph G is even if such a graph G satisfies that G∈Ck and η(L(G))=k+1, and the nullity set of Lk is {0,1,…,k,k+1} for any given k, where L(G) denotes the line graph of the graph G.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.