Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D-, C-, or B-hypergraph respectively, where B = C = D. D-hypergraph colorings are the classic hypergraph colorings which have been widely studied. The chromatic polynomial P(H;λ) of a mixed hypergraph H is the function that counts the number of proper λ-colorings, which are mappings. Recently, Walter published [15] some results concerning the chromatic polynomial of some non-uniform D-sunflower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunflowers and B-sunflowers. Some results of a new but related member of sunflowers are also presented

Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring ofthe vertex set X so that no D??hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D-, C-, or B-hypergraphrespectively where B = C = D. D-hypergraph colorings are the classichypergraph colorings which have been widely studied. The chro-matic polynomial P(H;) of a mixed hypergraph H is the function thatcounts the number of proper ??colorings, which are mappings f : X !f1; 2; : : : ; g. A sunfower (hypergraph) with l petals and a core S is a collection of sets e1; : : : ; el such that ei \ ej = S for all i 6= j. Recently, Walter published [14] some results concerning the chromatic polynomial of some non-uniform D-sunfower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunowers and B-sunowers. Some results of a new but related member of sunfowers are also presented

Graph Theory

This is the report on an Oberwolfach conference on graph theory, held 16-22 January 2005. There were three main components to the event: 5-minute presentations, lectures, and workshops. All participants were asked to give a 5-minute presentation of their interests on the first day, and subsequent days were divided into lectures and workshops. The latter ranged over many different topics, but the main three topics were: infinite graphs, topological methods and their use to prove theorems in graph theory, and Rota’s conjecture for matroids

Chromatic Polynomials of Some Mixed Hypergraphs

Motivated by a recent result of M. Walter [Electron. J. Comb. 16, No. 1, Research Paper R94, 16 p. (2009; Zbl 1186.05059)] concerning the chromatic polynomials of some hypergraphs, we present the chromatic polynomials of several (non-uniform) mixed hypergraphs. We use a recursive process for generating explicit formulae for linear mixed hypercacti and multi-bridge mixed hypergraphs using a decomposition of the underlying hypergraph into blocks, defined via chains. Further, using an algebra software package such as Maple, one can use the basic formulae and process demonstrated in this paper to generate the chromatic polynomials for any linear mixed hypercycle, unicyclic mixed hypercyle, mixed hypercactus and multi-bridge mixed hypergraph. We also give the chromatic polynomials of several examples in illustration of the process including the formulae for some mixed sunflowers

On extremal spectral radius of blow-up uniform hypergraphs

Let $G$ be an $r$-uniform hypergraph of order $t$ and $\rho(G)$ is the spectral radius of $\mathcal{A}(G)$, where $\mathcal{A}(G)$ is the adjacency tensor of $G$. A blow-up of $G$ respected to a positive integer vector $(n_{1}, n_{2},\ldots,n_{t})$, denoted by $G \circ (n_{1}, n_{2},\ldots,n_{t})$, is an $r$-uniform hypergraph obtained from $G$ by replacing each vertex $j$ of $G$ with a class of vertices $V_{j}$ of size $n_{j}\ge 1$ and if $\{j_{1},j_{2},\ldots,j_{r}\}\in E(G)$, then $\{v_{i_1},v_{i_2},\ldots,v_{i_r}\}\in E(H)$ for every $v_{i_{1}}\in V_{j_{1}}, v_{i_{2}}\in V_{j_{2}},\ldots, v_{i_{r}}\in V_{j_{r}}$. Let $\mathcal{B}_{n}(G)$ be the set of all the blow-ups of $G$ such that each $n_i\ge 1$ and $\sum_{i=1}^n n_i=n$. Let $K_{t}^{r}$ be the complete $r$-uniform hypergraph of order $t$, and let $SH(m,q,r)$ be the $r$-uniform sunflower hypergraph with $m$ petals and a kernel of size $r-q$ on $t$ vertices. For any $H\in \mathcal{B}_{n}(K_{t}^{r})$, we prove that $$\rho(K_{t}^{r}\circ(n-t+1,1,1,\ldots,1))\leq\rho(H)\leq \rho (T_{t}^{r}(n)),$$ with the left equality holds if and only if $H\cong K_{t}^{r}\circ(n-t+1,1,1,\ldots,1)$, and the right equality holds if and only if $H\cong T_{t}^{r}(n)$, where $T_{t}^{r}(n)$ is the complete $t$-partite $r$-uniform hypergraph of order $n$, with parts of size $\lfloor n / k\rfloor$ or $\lceil n / k \rceil$. For any $H\in \mathcal{B}_{n}(H(m,q,r))$, we determine the exact value of the spectral radius of $H$ and characterize the hypergraphs with maximum spectral radius and minimum spectral radius in $\mathcal{B}_{n}(H(m,q,r))$, respectively.Comment: 16 page

Coloring and constructing (hyper)graphs with restrictions

We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties. In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph. In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős). In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor. In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases. In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound