We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a $(5-varepsilon)$-approxi
mation, for any $varepsilon>0$. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.
Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.
Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, inclu
ding planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.
We prove that for any $varepsilon>0$, for any large enough $t$, there is a graph $G$ that admits no $K_t$-minor but admits a $(frac32-varepsilon)t$-colouring that is frozen with respect to Kempe changes, i.e. any two colour classes induce a connected
component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.
Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $chi(G)$, and the maximum is the so-cal
led Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $chi_c(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $chi_c(G)>chi(G)$. We prove that $chi(G)=chi_c(G)$ if $G$ is bipartite, and that $chi_c(G)leq 4$ if $G$ is subcubic.
A hereditary class of graphs $mathcal{G}$ is emph{$chi$-bounded} if there exists a function $f$ such that every graph $G in mathcal{G}$ satisfies $chi(G) leq f(omega(G))$, where $chi(G)$ and $omega(G)$ are the chromatic number and the clique number o
f $G$, respectively. As one of the first results about $chi$-bounded classes, Gy{a}rf{a}s proved in 1985 that if $G$ is $P_t$-free, i.e., does not contain a $t$-vertex path as an induced subgraph, then $chi(G) leq (t-1)^{omega(G)-1}$. In 2017, Chudnovsky, Scott, and Seymour proved that $C_{geq t}$-free graphs, i.e., graphs that exclude induced cycles with at least $t$ vertices, are $chi$-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that $P_{t-1}$-free graphs are in particular $C_{geq t}$-free. It remains a major open problem in the area whether for $C_{geq t}$-free, or at least $P_t$-free graphs $G$, the value of $chi(G)$ can be bounded from above by a polynomial function of $omega(G)$. We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every $t$ there exists a constant $c$ such that for and every $C_{geq t}$-free graph which does not contain $K_{ell,ell}$ as a subgraph, it holds that $chi(G) leq ell^{c}$.
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show the
ir applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.
We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing $2^{Omega(n^2)}$ $n$-vertex graphs as $nto infty$. This regime contains many classes of interest, for instance perfect graphs or comparability graph
s, for which we obtain an adjacency labelling scheme with labels of $n/4+o(n)$ bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of $n/4+o(n)$ bits per vertex and comparability labelling scheme for posets with labels of $n/4+o(n)$ bits per element. All these results are best possible, up to the lower order term.
We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the follo
wing three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Watanabe conjectured that every bipartite planar graph has an induced forest on at least five-eighths of its vertices, and in 2010, Kowalik, Luv{z}ar, and v{S}krekovski conjectured that every planar graph of girth at least five has an induced forest on at least seven-tenths of its vertices. We make progress toward the fractional generalization of the latter of these, by proving that every planar graph of girth at least five has fractional vertex-arboricity at most $2 - 1/324$.
We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $mathbb{S}^2$ in $mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probab
ility measures on $mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz number (the conjectured crossing number of $K_{n,n}$). In this paper we consider the intersection graphs associated with such random drawings. We prove that for any probability measures, the resulting random intersection graphs form a convergent graph sequence in the sense of graph limits. The edge density of the limiting graphon turns out to be independent of the two measures as long as they are antipodally symmetric. However, it is shown that the triangle densities behave differently. We examine a specific random model, blow-ups of antipodal drawings $D$ of $K_{4,4}$, and show that the triangle density in the corresponding crossing graphon depends on the angles between the great circles containing the edges in $D$ and can attain any value in the interval $bigl(frac{83}{12288}, frac{128}{12288}bigr)$.
