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Register a new userChromatic roots and limits of dense graphs
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In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.
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Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Delta(G)>n/3$ has chromatic index $Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K{u}hn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, K{u}hn and Osthus is affirmative.
Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph $G$, denoted by $chi_s(G)$, is the minimum number of colours needed in any strong edge colouring of $G$. A graph is said to be emph{chordless} if there is no cycle in the graph that has a chord. Faudree, Gyarfas, Schelp and Tuza~[The Strong Chromatic Index of Graphs, Ars Combin., 29B (1990), pp.~205--211] considered a particular subclass of chordless graphs, namely the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan~[Strong Chromatic Index of 2-degenerate Graphs, J. Graph Theory, 73(2) (2013), pp.~119--126] answered this question in the affirmative by proving that if $G$ is a chordless graph with maximum degree $Delta$, then $chi_s(G) leq 8Delta -6$. We improve this result by showing that for every chordless graph $G$ with maximum degree $Delta$, $chi_s(G)leq 3Delta$. This bound is tight up to to an additive constant.
A signed graph is a pair $(G, sigma)$, where $G$ is a graph and $sigma: E(G) to {+, -}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, sigma)$ a circular $r$-coloring of $(G, sigma)$ is an assignment $psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $sigma(e)=+$, then $psi(u)$ and $psi(v)$ have distance at least $1$, and if $sigma(e)=-$, then $psi(v)$ and the antipodal of $psi(u)$ have distance at least $1$. The circular chromatic number $chi_c(G, sigma)$ of a signed graph $(G, sigma)$ is the infimum of those $r$ for which $(G, sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $max{chi_c(G, sigma): sigma text{ is a signature of $G$}}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $chi_c(G, sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of M{a}v{c}ajov{a}, Raspaud, and v{S}koviera.
In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.
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