On Whitney's 2-isomorphism theorem for graphs

Let G and H be 2-connected 2-isomorphic graphs with n nodes. Whitney's 2-isomorphism theorem states that G may be transformed to a graph G* isomorphic to H by repeated application of a simple operation, which we will term “switching”. We present a proof of Whitney's theorem that is much shorter than the original one, using a graph decomposition by Tutte. The proof also establishes a surprisingly small upper bound, namely n-2, on the minimal number of switchings required to derive G* from G. The bound is sharp in the sense that for any integer N there exist graphs G and H with n ≥ N nodes for which the minimal number of switchings is n-2.     

Linear Ramsey numbers of sparse graphs

The Ramsey number R(G₁,G₂) of two graphs G₁ and G₂ is the least integer p so that either a graph G of order p contains a copy of G₁ or its complement G^c contains a copy of G₂. In 1973, Burr and Erd?s offered a total of $25 for settling the conjecture that there is a constant c = c(d) so that R(G,G)≤ c|V(G)| for all d‐degenerate graphs G, i.e., the Ramsey numbers grow linearly for d‐degenerate graphs. We show in this paper that the Ramsey numbers grow linearly for degenerate graphs versus some sparser graphs, arrangeable graphs, and crowns for example. This implies that the Ramsey numbers grow linearly for degenerate graphs versus graphs with bounded maximum degree, planar graphs, or graphs without containing any topological minor of a fixed clique, etc. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Planar Ramsey numbers for cycles

For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.     

Threshold functions for asymmetric Ramsey properties involving cycles

We consider the binomial random graph G_p and determine a sharp threshold function for the edge-Ramsey property for all l₁,…,l_r, where C^l denotes the cycle of length l. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 245–276, 1997     

Fault Tolerance of Cayley Graphs

It is a difficult problem in general to decide whether a Cayley graph Cay(G; S) is connected where G is an arbitrary finite group and S a subset of G. For example, testing primitivity of an element in a finite field is a special case of this problem but notoriously hard. In this paper, it is shown that if a Cayley graph Cay(G; S) is known to be connected then its fault tolerance can be determined in polynomial time in |S|log(|G|). This is accomplished by establishing a new structural result for Cayley graphs. This result also yields a simple proof of optimal fault tolerance for an infinite class of Cayley graphs, namely exchange graphs. We also use the proof technique for our structural result to give a new proof of a known result on quasiminimal graphs. Received March 10, 2006     

Induced Ramsey Numbers

   We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

A GAME-THEORETIC PROOF OF ANALYTIC RAMSEY THEOREM

We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.     

Mixed ramsey numbers: Chromatic numbers versus graphs

For positive integers n₁, n₂, …, n_I and graphs G_I+1, G_I+2, …, G_k, 1 ≤ / < k, the mixed Ramsey number _χ(n₁, …, n₁, G_I+1, …, G_k) is define as the least positive integer p such that for each factorization K_p = F₁⊕ … ⊕ F_⊕ F_I+1⊕ … ⊕ F_k, it it follows that χ(F_i) ≥ n_i for some i, 1 ? i ? l, or G_i is a subgraph of F_i for some i, l < i ? k. Formulas are presented for maxed Ramsey numbers in which the graphs G_I+1, G_I+2, …, G_k are connected, and in which k = I+1 and G_I+1 is arbitray.     

Improved boolean formulas for the Ramsey graphs

In the theory of the random graphs, there are properties of graphs such that almost all graphs satisfy the property, but there is no effective way to find examples of such graphs. By the well-known results of Razborov, for some of these properties, e.g., some Ramsey property, there are Boolean formulas in ACC representing the graphs satisfying the property and having exponential number of vertices with respect to the number of variables of the formula. Razborov's proof is based on a probabilistic distribution on formulas of n variables of size approximately nd² log d, where d is a polynomial in n, and depth 3 in the basis { ∧, ⊕} with the following property: The restriction of the formula randomly chosen from the distribution to any subset A of the Boolean cube {0, 1}ⁿ of size at most d has almost uniform distribution on the functions A → {0, 1}. We show a modified probabilistic distribution on Boolean formulas which is defined on formulas of size at most nd log² d and has the same property of the restrictions to sets of size at most d as the original one. This allows us to obtain formulas the complexity of which is a polynomial of a smaller degree in n than in Razborov's paper while the represented graphs satisfy the same properties.     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Graphs with the balas—uhry property

We characterize graphs H with the following property: Let G be a graph and F be a subgraph of G such that (i) each component of F is isomorphic to H or K₂, (ii) the order of F is maximum, and (iii) the number of H-components in F is minimum subject to (ii). Then a maximum matching of F is also a maximum matching of G. This result is motivated by an analogous property of fractional matchings discovered independently by J. P. Uhry and E. Balas.     

An upper bound for the total chromatic number of dense graphs

In this paper we consider those graphs that have maximum degree at least 1/k times their order, where k is a (small) positive integer. A result of Hajnal and Szemerédi concerning equitable vertex-colorings and an adaptation of the standard proof of Vizing's Theorem are used to show that if the maximum degree of a graph G satisfies Δ(G) ≥ |V(G)/k, then X″(G) ≤ Δ(G) + 2k + 1. This upper bound is an improvement on the currently available upper bounds for dense graphs having large order.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

G ∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology #

In 1997, M. Kontsevich proved the L _∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G _∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L _∞-formality.     

Some small ramsey numbers

In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs F₁, F₂ of graphs where F₁ has at most four points and F₂ has exactly five points. Exact results are listed for almost all of these pairs.     

Simple proof of the existence of restricted ramsey graphs by means of a partite construction

By means of a partite construction we present a short proof of the Galvin Ramsey property of the class of all finite graphs and of its strengthening proved in [5]. We also establish a generalization of those results. Further we show that for every positive integerm there exists a graphH which is Ramsey forK _m and does not contain two copies ofK _m with more than two vertices in common.     

Acyclic vertex coloring of graphs of maximum degree 5

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. For a family F of graphs, the acyclic chromatic number of F, denoted by a(F), is defined as the maximum a(G) over all the graphs G∈F. In this paper we show that a(F)=8 where F is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound.     

The Ramsey Numbers of Large cycles Versus Odd Wheels

For given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C_n, W_m) = 3n − 2 for odd m ≥ 5 and . Surahmat, Ioan Tomescu: Part of the work was done while the first and the last authors were visiting the School of Mathematical Sciences, Government College University, Lahore, Pakistan. Surahmat: Research partially support under TWAS, Trieste, Italy, RGA No: 06-018 RG/MATHS/AS–UNESCO FR: 3240144875.     

A note on Ki-perfect graphs

Ki-perfect graphs are a special instance of F - G perfect graphs, where F and G are fixed graphs with F a partial subgraph of G. Given S, a collection of G-subgraphs of graph K, an F - G cover of S is a set of T of F-subgraphs of K such that each subgraph in S contains as a subgraph a member of T. An F - G packing of S is a subcollection S′? S such that no two subgraphs in S′ have an F-subgraph in common. K is F - G perfect if for all such S, the minimum cardinality of an F - G cover of S equals the maximum cardinality of an F - G packing of S. Thus K_i-perfect graphs are precisely K_i-1 - K_i perfect graphs. We develop a hypergraph characterization of F - G perfect graphs that leads to an alternate proof of previous results on K_i-perfect graphs as well as to a characterization of F - G perfect graphs for other instances of F and G.