Smallest graphs with given generalized quaternion automorphism group

For , a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If , then such a graph has vertices and edges. In the special case when , a smallest graph has 16 vertices but 44 edges.     

Coloring (gem, co‐gem)‐free graphs

A gem is a graph that consists of a path on four vertices plus a vertex adjacent to all four vertices of the path. A co‐gem is the complement of a gem. We prove that every (gem, co‐gem)‐free graph G satisfies the inequality (a special case of a conjecture of Gyárfás) and the inequality (a special case of a conjecture of Reed). Moreover, we give an ‐time algorithm that computes the chromatic number of any (gem, co‐gem)‐free graph with n vertices, while the existing algorithm in the literature takes .     

Calculating Ramsey Numbers by Partitioning Colored Graphs

In this article we prove a new result about partitioning colored complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for , in every edge coloring of with the colors red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete ‐partite graph. When the coloring of is connected in red, we prove a stronger result—that it is possible to cover all the vertices with k red paths and a blue balanced complete ‐partite graph. Using these results we determine the Ramsey number of an n‐vertex path, , versus a balanced complete t‐partite graph on vertices, , whenever . We show that in this case , generalizing a result of Erd?s who proved the case of this result. We also determine the Ramsey number of a path versus the power of a path . We show that , solving a conjecture of Allen, Brightwell, and Skokan.     

Two Questions of Erdős on Hypergraphs above the Turán Threshold

For ordinary graphs it is known that any graph G with more edges than the Turán number of must contain several copies of , and a copy of , the complete graph on vertices with one missing edge. Erd?s asked if the same result is true for , the complete 3‐uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for . For the second property, that of containing a , we show that for the answer is negative for all large n as well, by proving that the Turán density of is greater than that of .     

Chasing a Fast Robber on Planar Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let denote the number of cops needed to capture the robber in a graph G in this variant, and let denote the treewidth of G. We show that if G is planar then , and there is a polynomial‐time constant‐factor approximation algorithm for computing . We also determine, up to constant factors, the value of of the Erd?s–Rényi random graph for all admissible values of p, and show that when the average degree is ω(1), is typically asymptotic to the domination number.     

Extensions of Fractional Precolorings Show Discontinuous Behavior

We study the following problem: given a real number k and an integer d, what is the smallest ε such that any fractional ‐precoloring of vertices at pairwise distances at least d of a fractionally k‐colorable graph can be extended to a fractional ‐coloring of the whole graph? The exact values of ε were known for and any d. We determine the exact values of ε for if , and if , and give upper bounds for if , and if . Surprisingly, ε viewed as a function of k is discontinuous for all those values of d.     

Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Let be an integer, be the set of vertices of degree at least 2k in a graph G , and be the set of vertices of degree at most in G . In 1963, Dirac and Erd?s proved that G contains k (vertex) disjoint cycles whenever . The main result of this article is that for , every graph G with containing at most t disjoint triangles and with contains k disjoint cycles. This yields that if and , then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with and contains k disjoint cycles.     

On the Structure of Graphs with Given Odd Girth and Large Minimum Degree

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd?s, and Sós implies that every n‐vertex graph with odd girth and minimum degree bigger than must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases and 3, we show that every n‐vertex graph with odd girth and minimum degree bigger than is homomorphic to the cycle of length . This is best possible in the sense that there are graphs with minimum degree and odd girth that are not homomorphic to the cycle of length . Similar results were obtained by Brandt and Ribe‐Baumann.     

Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph G with colors is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree admits a cyclic interval ‐coloring if for every vertex v the degree satisfies either or . We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for ‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all ‐biregular () graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.     

Smallest domination number and largest independence number of graphs and forests with given degree sequence

For a sequence d of nonnegative integers, let and be the sets of all graphs and forests with degree sequence d, respectively. Let , , , and where is the domination number and is the independence number of a graph G. Adapting results of Havel and Hakimi, Rao showed in 1979 that can be determined in polynomial time. We establish the existence of realizations with , and with and that have strong structural properties. This leads to an efficient algorithm to determine for every given degree sequence d with bounded entries as well as closed formulas for and .     

Hamilton Cycles,Minimum Degree,and Bipartite Holes

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large “bipartite hole” (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chvátal and Erd?s. In detail, an ‐bipartite‐hole in a graph G consists of two disjoint sets of vertices S and T with and such that there are no edges between S and T ; and is the maximum integer r such that G contains an ‐bipartite‐hole for every pair of nonnegative integers s and t with . Our central theorem is that a graph G with at least three vertices is Hamiltonian if its minimum degree is at least . From the proof we obtain a polynomial time algorithm that either finds a Hamilton cycle or a large bipartite hole. The theorem also yields a condition for the existence of k edge‐disjoint Hamilton cycles. We see that for dense random graphs , the probability of failing to contain many edge‐disjoint Hamilton cycles is . Finally, we discuss the complexity of calculating and approximating .     

An Asymptotic Version of the Multigraph 1‐Factorization Conjecture

We give a self‐contained proof that for all positive integers r and all , there is an integer such that for all any regular multigraph of order 2n with multiplicity at most r and degree at least is 1‐factorizable. This generalizes results of Perkovi? and Reed (Discrete Math 165/166 (1997), 567–578) and Plantholt and Tipnis (J London Math Soc 44 (1991), 393–400).     

Ramsey Results for Cycle Spectra

Let denote the set of lengths of cycles of a graph G of order n and let denote the complement of G. We show that if , then contains all odd ? with and all even ? with , where and denote the maximum odd and the maximum even integer in , respectively. From this we deduce that the set contains at least integers, which is sharp.     

Full subgraphs

Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define Erdős, Łuczak, and Spencer proved that for , In this article, we prove the following lower bound: for , Furthermore, we show that this is tight up to a multiplicative constant factor for infinitely many p near the elements of . In contrast, we show that for any n‐vertex graph G, either G or contains a full subgraph on vertices. Finally, we discuss full subgraphs of random and pseudo‐random graphs, and several open problems.     

Complete graph immersions and minimum degree

An immersion of a graph H in another graph G is a one‐to‐one mapping and a collection of edge‐disjoint paths in G, one for each edge of H, such that the path corresponding to the edge has endpoints and . The immersion is strong if the paths are internally disjoint from . We prove that every simple graph of minimum degree at least contains a strong immersion of the complete graph . This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al. Our result supports a conjecture of Lescure and Meyniel (also independently proposed by Abu‐Khzam and Langston), which is the analogue of famous Hadwiger’s conjecture for immersions and says that every graph without a ‐immersion is ‐colorable.     

The width of quadrangulations of the projective plane

We show that every 4‐chromatic graph on n vertices, with no two vertex‐disjoint odd cycles, has an odd cycle of length at most . Let G be a nonbipartite quadrangulation of the projective plane on n vertices. Our result immediately implies that G has edge‐width at most , which is sharp for infinitely many values of n. We also show that G has face‐width (equivalently, contains an odd cycle transversal of cardinality) at most , which is a constant away from the optimal; we prove a lower bound of . Finally, we show that G has an odd cycle transversal of size at most inducing a single edge, where Δ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.     

The decycling number and maximum genus of cubic graphs

Let and denote the minimum size of a decycling set and maximum genus of a graph G, respectively. For a connected cubic graph G of order n, it is shown that . Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k‐regular graph G is , where c and are the number of components of and the number of edges in , respectively. Therefore, S is minimum if and only if is minimum. As an application, this leads to a lower bound for of a k‐regular graph G. In many cases this bound may be sharp.     

Low Polynomial Exclusion of Planar Graph Patterns

The celebrated grid exclusion theorem states that for every h‐vertex planar graph H , there is a constant such that if a graph G does not contain H as a minor then G has treewidth at most . We are looking for patterns of H where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel , the double wheel , any graph of pathwidth at most 2 , and the yurt graph .     

Quasi‐carousel tournaments

A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W₄ and L₄, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments  with  is called almost balanced if all but  vertices of  have outdegree . In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W₄ and L₄ in  goes to zero as n goes to infinity.