Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $C_n$ denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joining two cycles $C_6$ by a path $P_{n-12}$ with its two leaves. Let $\mathscr{B}_n$ denote the class of all bipartite bicyclic graphs but not the graph $R_{a,b}$, which is obtained from joining two cycles $C_a$ and $C_b$ ($a, b\geq 10$ and $a \equiv b\equiv 2\, (\,\textmd{mod}\, 4)$) by an edge. In [I. Gutman, D. Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computer experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and Vidovi\'{c} conjectured that the bicyclic graph with maximal energy is $P^{6,6}_n$, for $n=14$ and $n\geq 16$. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and Zhang showed that the conjecture is true for graphs in the class $\mathscr{B}_n$. However, they could not determine which of the two graphs $R_{a,b}$ and $P^{6,6}_n$ has the maximal value of energy. In [B. Furtula, S. Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy, {\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations up to $a+b=50$ were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of $P^{6,6}_n$ is larger than that of $R_{a,b}$, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.Comment: 9 page

Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that $G$ is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of $G$, then we can extend these path systems into an approximate decomposition of $G$ into Hamilton cycles (or perfect matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the third paper. We have now combined this series into a single publication [arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2 figure

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201