Alon-Tarsi Number of Some Complete Multipartite Graphs

The Alon-Tarsi number of a polynomial is a parameter related to the exponents of its monomials. For graphs, their Alon-Tarsi number is the Alon-Tarsi number of their graph polynomials. As such, it provides an upper bound on their choice and online choice numbers. In this paper, we obtain the Alon-Tarsi number of some complete multipartite graphs and line graphs of some complete graphs of even order.Comment: 4 page

Towards on-line Ohba's conjecture

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 \chi(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead's result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le \chi(G)+\sqrt{\chi(G)}$ then its on-line choice number equals chromatic number.Comment: new abstract and introductio

On the Existence of S-graphs

We answer in the affirmative a question posed by S. Al-Addasi and H. Al-Ezeh in [Int. J. Math. Math. Sci. 2008, Article ID 468583, 11 p. (2008; Zbl 1161.05316)] on the existence of symmetric diametrical bipartite graphs of diameter 4. Bipartite symmetric diametrical graphs are calls S-graphs by some authors and diametrical graphs have also been studied by other authors using different terminology, such as self-centered unique eccentric point graphs. We include a brief survey of some of this literature and advertise that the existence question was answered by A. Berman and A. Kotzig [Ann. Discrete Math. 8, 37–42 (1980; Zbl 0446.05025)], along with a study of different isomorphism classes of these graphs using a (1,-1)-matrix representation which includes the well-known Hadamard matrices. Our presentation focuses on a neighborhood characterization of S-graphs and we conclude our survey with a beautiful version of this characterization known to T. N. Janakiraman [Discrete Math. 126, No. 1–3, 411–414 (1994; Zbl 0792.05117)]]]> 2012 English http://libres.uncg.edu/ir/ecsu/f/Allagan-S-graphs_AHJS.pdf oai:libres.uncg.edu/38718 2023-02-19T20:48:38Z ECSU Hall numbers of some complete k-partite graphs Allagan , Julian A. D. NC DOCKS at Elizabeth City State University <![CDATA[The Hall number is a graph parameter closely related to the choice number. Here it is shown that the Hall numbers of the complete multipartite graphs K(m,2,?,2), m=2, are equal to their choice numbers

The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k})$. Also, for every $\mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So

A Short Note on Undirected Fitch Graphs

The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information

Transversal designs and induced decompositions of graphs

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $\Omega(n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-decomposition