Graphs With 3n−6 Edges Not Containing A Subdivision Of K 5

We determine all graphs on n ≥ 3 vertices with 3n-6 edges which do not contain a subdivision of K₅. These are exactly the graphs which one gets from any number of disjoint maximal planar graphs by successively pasting along triangles.     

Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices

In this paper, we describe the generation of all nonorientable triangular embeddings of the complete graphs K₁₂ and K₁₃. (The 59 nonisomorphic orientable triangular embeddings of K₁₂ were found in 1996 by Altshuler, Bokowski, and Schuchert, and K₁₃ has no orientable triangular embeddings.) There are 182,200 nonisomorphic nonorientable triangular embeddings for K₁₂, and 243,088,286 for K₁₃. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of K_n. We mention an application of our results to flexibility of embedded graphs. © 2005 Wiley Periodicals, Inc. J Combin Designs     

The crossing number of K11 is 100

The crossing number of K_n is known for n ? 10. We develop several simple counting properties that we shall exploit in showing by computer that cr(K₁₁ = 100, which implies that cr(K₁₂) = 150. We also determine the numbers of non‐isomorphic optimal drawings of K₉ and K₁₀. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 128–134, 2007     

The Crossing Number of Cartesian Products of Complete Bipartite Graphs K 2, m with Paths P n

Most results on the crossing number of a graph focus on the special graphs, such as Cartesian products of small graphs with paths P_n, cycles C_n or stars S_n. In this paper, we extend the results to Cartesian products of complete bipartite graphs K_2,m with paths P_n for arbitrary m ≥ 2 and n ≥ 1. Supported by the NSFC (No. 10771062) and the program for New Century Excellent Talents in University.     

The crossing number of K3, n in a surface

In this article, we show that the crossing number of K_3,n in a surface with Euler genus ϵ is ⌊n/(2ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌊n/(2ϵ + 2)⌋}). This generalizes a result of Guy and Jenkyns, who obtained this result for the torus. © 1996 John Wiley & Sons, Inc.     

Triangular embeddings of K((i-2) n,n,…,n)

For complete i-partite graphs of the form K(n₁, n, n, …, n) the largest value of n₁ that allows the graph to be triangularly-embedded into a surface is (i-2)n. In this paper the author constructs triangular embeddings into surfaces of some complete partite graphs of the form K((i-2)n, n, …, n). The embeddings are exhibited using embedding schemes but the surfaces into which K((i-2)n, n, …, n) are triangularly embedded can be seen to be particularly nice branched covers of a surface into which K(i-2, 1, 1,…,1) is triangularly embedded.     

Do 3n − 5 edges force a subdivision of K5?

A conjecture of Dirac states that every simple graph with n vertices and 3n ? 5 edges must contain a subdivision of K₅. We prove that a topologically minimal counterexample is 5-connected, and that no minor-minimal counterexample contains K₄ – e. Consequently, Dirac's conjecture holds for all graphs that can be embedded in a surface with Euler characteristic at least ? 2.     

Euler Tours of Maximum Girth in K2 n +1 and K2 n ,2 n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g^E(G)= maxg(T) where the maximum is taken over all Euler tours of G.We prove that g^E(K_2n,2n)=4n–4 and 2n–3g^E(K_2n+1)2n–1 for any n2. We also show that g^E(K₇)=4. We use these results to prove the following:1)The graph K_2n,2n can be decomposed into edge disjoint paths of length k if and only if k4n–1 and the number of edges in K_2n,2n is divisible by k.2)The graph K_2n+1 can be decomposed into edge disjoint paths of length k if and only if k2n and the number edges in K_2n+1 is divisible by k.     

On the Crossing Numbers of Products of Stars and Graphs of Order Five

 There are several known exact results on the crossing numbers of Cartesian products of paths or cycles with “small” graphs. In this paper we extend these results to the Cartesian products of two specific 5-vertex graphs with the star K _{1, n} . In addition, we give the crossing number of the graph obtained by adding two edges to the graph K _{1,4, n} in such a way that these new edges join a vertex of degree n+1 of the graph K _{1,4, n} with two its vertices of the same degree. Received: December 8, 1997 Final version received: August 14, 1998     

Edge Colouring of K2n with Spanning Star-Forests Receiving Distinct Colours

 Let S _ni be a star of size n _i and let S=S _n1∪…∪S _nk≠S _2n−3∪S ₁ or S ₂∪S ₂ be a spanning star-forest of the complete graph K _2n. We prove that K _2n has a proper (2n−1)-edge-colouring such that all the edges of S receive distinct colours. This result is very useful in the study of total-colourings of graphs. Received: March 8, 1995 / Revised: May 16, 1997     

On the crossing numbers of Cartesian products with trees

Zip product was recently used in a note establishing the crossing number of the Cartesian product K₁,n □ P_m. In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 287–300, 2007     

Multicolor bipartite Ramsey number of C4 and large Kn,n

Let br_k(C₄;K_{n, n}) be the smallest N such that if all edges of K_{N, N} are colored by k + 1 colors, then there is a monochromatic C₄ in one of the first k colors or a monochromatic K_{n, n} in the last color. It is shown that br_k(C₄;K_{n, n}) = Θ(n²/log²n) for k?3, and br₂(C₄;K_{n, n})≥c(n n/log²n)² for large n. The main part of the proof is an algorithm to bound the number of large K_{n, n} in quasi‐random graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 47‐54, 2011     

Achromatic Number of K 5 × K n for Small n

The achromatic number of a graph G is the maximum number of colours in a proper vertex colouring of G such that for any two distinct colours there is an edge of G incident with vertices of those two colours. We determine the achromatic number of the Cartesian product of K ₅ and K _n for all n ≤ 24.     

The number of K m,m -free graphs

A graph is called H-free if it contains no copy of H. Denote by f _n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f _n (H) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and R?dl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log₂ f _n (H) is not known. We prove that f _n (K _m,m ) ≤ 2 ^O (n ^2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K _m,m ) is conjectured to be Θ(n ^2−1/m ). Our method also yields a bound on the number of K _m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K _3,3-free graphs of order n have more than 1/20·ex(n,K _3,3) edges.     

All cycle‐complete graph Ramsey numbers r(Cm,K6)

The cycle‐complete graph Ramsey number r(C_m, K_n) is the smallest integer N such that every graph G of order N contains a cycle C_m on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erd?s, Faudree, Rousseau and Schelp that r(C_m, K_n) = (m ? 1) (n ? 1) + 1 for all m ≥ n ≥ 3 (except r(C₃, K₃) = 6). This conjecture holds for 3 ≤ n ≤ 5. In this paper we will present a proof for n = 6 and for all n ≥ 7 with m ≥ n² ? 2n. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 251–260, 2003     

Recursive constructions for triangulations

Three recursive constructions are presented; two deal with embeddings of complete graphs and one with embeddings of complete tripartite graphs. All three facilitate the construction of 2) non‐isomorphic face 2‐colourable triangulations of K_n and K_n,n,n in orientable and non‐orientable surfaces for values of n lying in certain residue classes and for appropriate constants a. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 87–107, 2002     

On the existence of rainbows in 1-factorizations of K2n

A 1-factor of a graph G = (V, E) is a collection of disjoint edges which contain all the vertices of V. Given a 2n - 1 edge coloring of K_2n, n ≥ 3, we prove there exists a 1-factor of K_2n whose edges have distinct colors. Such a 1-factor is called a “Rainbow.” © 1998 John Wiley & Sons, Inc. J Combin Designs 6:1–20, 1998     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

A multipartite Ramsey number for odd cycles

In this article we study multipartite Ramsey numbers for odd cycles. Our main result is the proof that a conjecture of Gyárfás et al. (J Graph Theory 61 (2009), 12–21), holds for graphs with a large enough number of vertices. Precisely, there exists n₀ such that if n?n₀ is a positive odd integer then any two‐coloring of the edges of the complete five‐partite graph K_{(n ? 1)/2, (n ? 1)/2, (n ? 1)/2, (n ? 1)/2, 1} contains a monochromatic cycle of length n. © 2011 Wiley Periodicals, Inc. J Graph Theory