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 .     

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.     

Saturation numbers for families of graph subdivisions

For a family of graphs, a graph G is ‐saturated if G contains no member of as a subgraph, but for any edge in , contains some member of as a subgraph. The minimum number of edges in an ‐saturated graph of order n is denoted . A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let denote the family of H‐subdivisions, including H itself. In this paper, we study when H is one of or , obtaining several exact results and bounds. In particular, we determine exactly for and show for n sufficiently large that there exists a constant such that . For we show that will suffice, and that this can be improved slightly depending on the value of . We also give an upper bound on for all t and show that . This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge‐maximal graphs without a K₅‐minor have at least edges.     

Dominating sets inducing large components in maximal outerplanar graphs

For a maximal outerplanar graph G of order n at least three, Matheson and Tarjan showed that G has domination number at most . Similarly, for a maximal outerplanar graph G of order n at least five, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that G has total domination number at most unless G is isomorphic to one of two exceptional graphs of order 12. We present a unified proof of a common generalization of these two results. For every positive integer k, we specify a set of graphs of order at least and at most such that every maximal outerplanar graph G of order n at least that does not belong to has a dominating set D of order at most such that every component of the subgraph of G induced by D has order at least k.     

List Coloring with a Bounded Palette

Král' and Sgall (J Graph Theory 49(3) (2005), 177–186) introduced a refinement of list coloring where every color list must be subset to one predetermined palette of colors. We call this ‐choosability when the palette is of size at most ? and the lists must be of size at least k . They showed that, for any integer , there is an integer , satisfying as , such that, if a graph is ‐choosable, then it is C‐choosable, and asked if C is required to be exponential in k . We demonstrate it must satisfy . For an integer , if is the least integer such that a graph is ‐choosable if it is ‐choosable, then we more generally supply a lower bound on , one that is super‐polynomial in k if , by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on that improve on earlier bounds if .     

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. ).     

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.     

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 .     

Packing Countably Many Branchings with Prescribed Root‐Sets in Infinite Digraphs

We generalize an unpublished result of C. Thomassen. Let be a digraph and let be a multiset of subsets of V in such a way that any backward‐infinite path in D meets all the sets . We show that if all is simultaneously reachable from the sets by edge‐disjoint paths, then there exists a system of edge‐disjoint spanning branchings in D where the root‐set of is .     

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.     

More on the Bipartite Decomposition of Random Graphs

For a graph , let denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G , , where is the maximum size of an independent set of G . Erd?s conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph , with high probability, that is with probability that tends to 1 as n tends to infinity. The first author showed that this is slightly false, proving that for most values of n tending to infinity and for , with high probability. We prove a stronger bound: there exists an absolute constant so that with high probability.     

A relaxation of the strong Bordeaux Conjecture

Let be k nonnegative integers. A graph G is ‐colorable if the vertex set can be partitioned into k sets , such that the subgraph , induced by , has maximum degree at most for . Let denote the family of plane graphs with neither adjacent 3‐cycles nor 5‐cycles. Borodin and Raspaud (2003) conjectured that each graph in is (0, 0, 0)‐colorable (which was disproved very recently). In this article, we prove that each graph in is (1, 1, 0)‐colorable, which improves the results by Xu (2009) and Liu‐Li‐Yu (2016).     

Large Induced Forests in Graphs

In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least . The second is that every connected triangle‐free graph of order n and size m has an induced forest of order at least . This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound of Alon–Mubayi–Thomas for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least . The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least , where was conjectured to be the best lower bound by Kowalik, Lu?ar, and ?krekovski.     

Strong Chromatic Index of Sparse Graphs

A coloring of the edges of a graph G is strong if each color class is an induced matching of G. The strong chromatic index of G, denoted by , is the least number of colors in a strong edge coloring of G. Chang and Narayanan (J Graph Theory 73(2) (2013), 119–126) proved recently that for a 2‐degenerate graph G. They also conjectured that for any k‐degenerate graph G there is a linear bound , where c is an absolute constant. This conjecture is confirmed by the following three papers: in (G. Yu, Graphs Combin 31 (2015), 1815–1818), Yu showed that . In (M. Debski, J. Grytczuk, M. Sleszynska‐Nowak, Inf Process Lett 115(2) (2015), 326–330), D?bski, Grytczuk, and ?leszyńska‐Nowak showed that . In (T. Wang, Discrete Math 330(6) (2014), 17–19), Wang proved that . If G is a partial k‐tree, in (M. Debski, J. Grytczuk, M. Sleszynska‐Nowak, Inf Process Lett 115(2) (2015), 326–330), it is proven that . Let be the line graph of a graph G, and let be the square of the line graph . Then . We prove that if a graph G has an orientation with maximum out‐degree k, then has coloring number at most . If G is a k‐tree, then has coloring number at most . As a consequence, a graph with has , and a k‐tree G has .     

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.     

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.     

Sufficient conditions for the existence of a path‐factor which are related to odd components

In this article, we are concerned with sufficient conditions for the existence of a ‐factor. We prove that for , there exists such that if a graph G satisfies for all , then G has a ‐factor, where is the number of components C of with . On the other hand, we construct infinitely many graphs G having no ‐factor such that for all .