This result completes work of Balogh, Hu, Lidický, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs. ]]>

Our contribution here is to give a new class of tripartite common graphs. The first example class is so-called triangle-trees, which generalises two theorems by Sidorenko and answers a question by Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree T, there exists a triangle-tree such that the graph obtained by adding T as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most five vertices give a common graph.

]]>The main studied question is whether there exists a universal constant ϵ>0 such that any graph G in some graph class C satisfies at least ϵ proportion of the requests. More formally, for k>0 the goal is to prove that for any graph G∈C on vertex set V, with any list assignment L of size k for each vertex, and for every R⊆V and a request vector (r(v):v∈R, r(v)∈L(v)), there exists an L-coloring of G satisfying at least ϵ|R| requests. If this is true, then C is called ϵ-flexible for lists of size k.

Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where R=V. We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists ϵ(b)>0 so that the class of planar graphs without K4,C5,C6,C7,Bb is weakly ϵ(b)-flexible for lists of size 4 (here Kn, Cn and Bn are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K4,C5,C6,C7,B5 is ϵ-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

]]>The set of all real symmetric matrices having nonzero off-diagonal entries exactly where the graph G has edges is denoted by S( G). Given a graph G, the problem of characterizing the possible spectra of B, such that B. S( G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees.

The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S( G). Recent work on generalized Laplacians and Colin de Verdiere matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1-matrix or a generalized Laplacian. ]]>