Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .     

Minimum Vertex Degree Threshold for ‐tiling*

We prove that the vertex degree threshold for tiling (the 3‐uniform hypergraph with four vertices and two triples) in a 3‐uniform hypergraph on vertices is , where if and otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.     

Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles,Matchings, or Trees

We seek the maximum number of colors in an edge‐coloring of the complete graph not having t edge‐disjoint rainbow spanning subgraphs of specified types. Let , , and denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove for and for . We prove for and for . We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than q edges at a vertex.     

Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

We construct a face two‐colourable, blue and green say, embedding of the complete graph in a nonorientable surface in which there are blue faces each of which have a hamilton cycle as their facial walk and green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of , for all and . Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph , for where and .     

Total Vertex Irregularity Strength of Dense Graphs

Consider a graph of minimum degree δ and order n. Its total vertex irregularity strength is the smallest integer k for which one can find a weighting such that for every pair of vertices of G. We prove that the total vertex irregularity strength of graphs with is bounded from above by . One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

Interval Minors of Complete Bipartite Graphs

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley–Wilf limits. We investigate the maximum number of edges in ‐interval minor‐free bipartite graphs. We determine exact values when and describe the extremal graphs. For , lower and upper bounds are given and the structure of ‐interval minor‐free graphs is studied.     

Cycles Avoiding a Color in Colorful Graphs

The Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .     

Spanning Trees with Vertices Having Large Degrees

Let G be a connected simple graph, and let f be a mapping from to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least . We show that if for any nonempty subset , then a connected graph G has a spanning tree such that for all , where is the set of neighbors v of vertices in S with , , and is the degree of x in T. This is an improvement of several results, and the condition is best possible.     

Improper Coloring of Sparse Graphs with a Given Girth,II: Constructions

A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all .     

Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph

We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ).     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

Two‐sided Group Digraphs and Graphs

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.     

On the Circular Chromatic Number of Graph Powers

This article intends to study some functors from the category of graphs to itself such that, for any graph G, the circular chromatic number of is determined by that of G. In this regard, we investigate some coloring properties of graph powers. We show that provided that . As a consequence, we show that if , then . In particular, and has no subgraph with circular chromatic number equal to . This provides a negative answer to a question asked in (X. Zhu, Discrete Math, 229(1–3) (2001), 371–410). Moreover, we investigate the nth multichromatic number of subdivision graphs. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that .     

On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Let T be a tournament of order n and be the number of cycles of length m in T. For and odd n, the maximum of is achieved for any regular tournament of order n (M. G. Kendall and B. Babington Smith, 1940), and in the case it is attained only for the unique regular locally transitive tournament RLT_n of order n (U. Colombo, 1964). A lower bound was also obtained for in the class of regular tournaments of order n (A. Kotzig, 1968). This bound is attained if and only if T is doubly regular (when ) or nearly doubly regular (when ) (B. Alspach and C. Tabib, 1982). In the present article, we show that for any regular tournament T of order n, the equality holds. This allows us to reduce the case to the case In turn, the pure spectral expression for obtained in the class implies that for a regular tournament T of order the inequality holds, with equality if and only if T is doubly regular or T is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of c₆(RLT_n) and conjecture that this value coincides with the minimum number of 6‐cycles in the class for each odd      

Face 2‐Colorable Embeddings with Faces of Specified Lengths

Suppose and are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2‐colorable planar embedding in which the faces of one color have boundary lengths and the faces of the other color have boundary lengths . Such a graph is said to have a planar ‐biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths and also into t cycles of lengths . Such a graph is said to be ‐decomposable.     

An Extremal Property of Turán Graphs,II

Let denote Turán's graph—the complete 2‐partite graph on n vertices with partition sizes as equal as possible. We show that for all , the graph has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges.     

The Images of the Clique Operator and Its Square are Different

Let be the class of all graphs and K be the clique operator. The validity of the equality has been an open question for several years. A graph in but not in is exhibited here.     

Chordal 2‐Connected Graphs and Spanning Trees

We present a transformation on a chordal 2‐connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with , the threshold graph having n vertices and m edges that consists of an ‐clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2‐connected chordal graphs on n vertices and m edges.     

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let denote the number of convex cycles of a simple graph G of order n, size m, and girth . It is proved that and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.