Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.     

Vertex colorings without isolates

Call a vertex of a vertex-colored simple graph isolated if all its neighbors have colors other than its own. A. J. Goldman has asked: When is it possible to color b vertices of a graph black and the remaining w vertices white so that no vertex is isolated? We prove (1) if G is connected and has minimum degree 2, it is always possible unless b or w is 1; (2) if G is 2-connected, then for any pair (b, w) there is a coloring in which both monochromatic subgraphs are connected; (3) if G has vertices of degree 1, a necessary condition for a (b, w) coloring without isolates to exist is that there be a solution to a certain knapsack inequality. Next, statements generalizing (1) and (2) to n colors are presented, and current knowledge about their truth is discussed. Then various refinements of (1) and (3), more complicated to state and prove, are given. For instance, with the hypotheses of (1) at least one of the monochromatic subgraphs may be chosen to be connected. Also, the necessary knapsack inequality of (3) is, in most cases, sufficient. Throughout, some consideration is given to the algorithmic complexity of coloring (if possible) without isolates. For most graphs which might arise in practice there is an efficient algorithm for the 2-color problem. However, for arbitrary graphs the 2-(or more) color problem is NP-complete.     

A characterization of locally finite vertex-transitive graphs

A locally finite graph G with no isolated vertices is vertex-transitive if and only if all its vertex-deleted subgraphs G-v are isomorphic.     

Generalized complementation

Suppose the edges of a complete graph are colored using three colors, without forming any trichromatic triangles. We study those properties which must hold in at least one of the three monochromatic subgraphs and those whose truth in any two implies truth in the third. We also relate these phenomena to the two-color (complementation) case.     

Distance-residual subgraphs

For a connected finite graph G and a subset V₀ of its vertex set, a distance-residual subgraph is a subgraph induced on the set of vertices at the maximal distance from V₀. Some properties and examples of distance-residual subgraphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are presented. The relations between the distance-residual subgraphs of product graphs and their factors are explored.     

A distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} and its automorphisms

Possible orders and subgraphs of the fixed points of a distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} are found. It is shown that such a graph is not vertex-transitive.     

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?     

Partitioning infinite trees

A graph G is bisectable if its edges can be colored by two colors so that the resulting monochromatic subgraphs are isomorphic. We show that any infinite tree of maximum degree Δ with infinitely many vertices of degree at least Δ −1 is bisectable as is any infinite tree of maximum degree Δ ≤ 4. Further, it is proved that every infinite tree T of finite maximum degree contains a finite subset E of its edges so that the graph T − E is bisectable. To measure how “far” a graph G is from being bisectable, we define c(G) to be the smallest number k > 1 so that there is a coloring of the edges of G by k colors with the property that any two monochromatic subgraphs are isomorphic. An upper bound on c(G), which is in a sense best possible, is presented. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 113–127, 2000     

On construction of vertex-transitive partitions of the <Emphasis Type="Italic">n</Emphasis>-cube into perfect codes

Two methods are presented to construct some vertex-transitive and 2-transitive partitions of the n-cube into perfect codes. Some lower bounds are given on the number of transitive, vertex-transitive, and 2-transitive partitions of the n-cube into perfect codes.     

Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either. Supported by NSFC, PCSIRT and the “973” program.     

Ramsey‐type results for Gallai colorings

A Gallai‐coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai‐colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai‐colorings extend 2‐colorings of the edges of complete graphs. They actually turn out to be close to 2‐colorings—without being trivial extensions. Here, we give a method to extend some results on 2‐colorings to Gallai‐colorings, among them known and new, easy and difficult results. The method works for Gallai‐extendible families that include, for example, double stars and graphs of diameter at most d for 2?d, or complete bipartite graphs. It follows that every Gallai‐colored K_n contains a monochromatic double star with at least 3n+ 1/4 vertices, a monochromatic complete bipartite graph on at least n/2 vertices, monochromatic subgraphs of diameter two with at least 3n/4 vertices, etc. The generalizations are not automatic though, for instance, a Gallai‐colored complete graph does not necessarily contain a monochromatic star on n/2 vertices. It turns out that the extension is possible for graph classes closed under a simple operation called equalization. We also investigate Ramsey numbers of graphs in Gallai‐colorings with a given number of colors. For any graph H let RG(r, H) be the minimum m such that in every Gallai‐coloring of K_m with r colors, there is a monochromatic copy of H. We show that for fixed H, RG (r, H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if H is a star (and we determine its value). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 233–243, 2010     

Size of monochromatic components in local edge colorings

An edge coloring of a graph is a local r coloring if the edges incident to any vertex are colored with at most r distinct colors. We determine the size of the largest monochromatic component that must occur in any local r coloring of a complete graph or a complete bipartite graph.     

Structure of vertex-transitive graphs

Every vertex-transitive graph has a characteristic structure. The specific details of structure of a vertex-transitive graph are closely related to the configuration of the parts of any minimum separating set of that graph. It is shown for a vertex-transitive graph G that (i) the number n of isomorphic atomic parts admitted by any minimum separating set S is unique for that G; (ii) G is then isomorphic to a disjoint union of m(≥2) copies of such a set of n atomic parts together with some additional edges joining them; (iii) any minimum separating set S of G consists of the vertex set of the union of some k(≥1) copies of the set of n atomic parts; (iv) at most one nonatomic part will be admitted in conjunction with one or more atomic parts by any minimum separating set of G.This configuration of structure extends to nonatomic parts. A vertex-transitive graph G may contain a not necessarily unique minimum separating set S which admits t(≥3) nonatomic parts. Then it is shown that (i) some (t ? 1)-set of these nonatomic parts are pairwise isomorphic; (ii) if the remaining nonatomic part is nonisomorphic to the others then it contains more vertices than the other parts; (iii) G is isomorphic to a disjoint union of d(≥2) copies of the set of q(≥t ? 1) isomorphic nonatomic parts together with some additional edges joining them; (iv) a minimum separating set S consists of the vertex set of the union of some y(≥1) copies of the set of q isomorphic nonatomic parts. For the case of t = 2 non-atomic parts admitted by a minimum separating set S, counterexamples of (iii) and (iv) are given.     

Chromatically Supremal Decompositions of Graphs

If G is a graph, a G-decomposition of a host graph H is a partition of the edges of H into subgraphs of H which are isomorphic to G. The chromatic index of a G-decomposition of H is the minimum number of colors required to color the parts of the decomposition so that parts which share a common node get different colors. We establish an upper bound on the chromatic index and characterize those decompositions which achieve it. The structurally most interesting of the decompositions with maximal chromatic index are associated with (v, k, 1)-designs.     

Hamiltonicity of 3-Arc Graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. We prove that any connected 3-arc graph is hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are hamiltonian. As a corollary we obtain that any vertex-transitive graph which is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three must be hamiltonian. This confirms the conjecture, for this family of vertex-transitive graphs, that all vertex-transitive graphs with finitely many exceptions are hamiltonian. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.     

Cubic edge-transitive graphs of order 2p3

Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215) that a regular edge-transitive graph of order 2p or 2p² is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that, with the exception of the Gray graph on 54 vertices, every cubic edge-transitive graph of order 2p³ is vertex-transitive.     

Coloring arcs of convex sets

In this note we consider Ramsey-type problems on graphs whose vertices are represented by the vertices of a convex polygon in the Euclidean plane. The edges of the graph are represented by the segments between the points of the polygon. The edges are arbitrarily colored by a fixed number of colors and the problem is to decide whether there exist monochromatic subgraphs of certain types satisfying some geometric conditions. We will give lower and upper bounds for these geometric Ramsey numbers for certain paths and cycles and also some exact values. It turns out that the particular type of the embedding is crucial for the growth rate of the corresponding geometric Ramsey numbers. In particular, the Ramsey numbers for crossing 4-cycles and t colors grow quadratically in t, while for convex 4-cycles they grow at least exponentially.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

Monochromatic Cycle Partitions in Local Edge Colorings

An edge coloring of a graph is said to be an r‐local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomassé, we prove that the vertex set of any 2‐locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number r, we show that the vertex set of any r‐locally colored complete graph may be partitioned into disjoint monochromatic cycles. This generalizes a result of Erd?s, Gyárfás, and Pyber.