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Register a new userOn rigidity, orientability and cores of random graphs with sliders
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Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G in mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.
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As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.
A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let $H_k(n,m)$ be a random $k$-uniform hypergraph on $n$ vertices formed by picking $m$ edges uniformly, independently and with replacement. It is easy to show that if $r geq r_c = 2^{k-1} ln 2 - (ln 2) /2$, then with high probability $H_k(n,m=rn)$ is not 2-colorable. We complement this observation by proving that if $r leq r_c - 1$ then with high probability $H_k(n,m=rn)$ is 2-colorable.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0rightarrow v_1rightarrow cdotsrightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_irightarrow v_j$ is an arc for all $0leq i<jleq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via ErdH{o}s theorem by Halld{o}rsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph $K(8,3)$ is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Grotzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvatal graph) are not semi-transitive. Hence, the Grotzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph $C(13;1,5)$ on 13 vertices) and dense ones (Tofts graphs). Finally, we show that each $4$-regular circulant graph (possibly containing triangles) is semi-transitive.
We consider the class of semi-transitively orientable graphs, which is a much larger class of graphs compared to transitively orientable graphs, in other words, comparability graphs. Ever since the concept of a semi-transitive orientation was defined as a crucial ingredient of the characterization of alternation graphs, also knownas word-representable graphs, it has sparked independent interest. In this paper, we investigate graph operations and graph products that preserve semitransitive orientability of graphs. The main theme of this paper is to determine which graph operations satisfy the following statement: if a graph operation is possible on a semitransitively orientable graph, then the same graph operation can be executed on the graph while preserving the semi-transitive orientability. We were able to prove that this statement is true for edge-deletions, edge-additions, and edge-liftings. Moreover, for all three graph operations,we showthat the initial semi-transitive orientation can be extended to the new graph obtained by the graph operation. Also, Kitaev and Lozin explicitly asked if certain graph products preserve the semitransitive orientability. We answer their question in the negative for the tensor product, lexicographic product, and strong product.We also push the investigation further and initiate the study of sufficient conditions that guarantee a certain graph operation to preserve the semi-transitive orientability.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0rightarrow v_1rightarrow cdotsrightarrow v_k$ either there is no edge between $v_0$ and $v_k$, or $v_irightarrow v_j$ is an edge for all $0leq i<jleq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs include several important classes of graphs such as 3-colorable graphs, comparability graphs, and circle graphs, and they are precisely the class of word-representable graphs studied extensively in the literature. In this paper, we study semi-transitive orientability of the celebrated Kneser graph $K(n,k)$, which is the graph whose vertices correspond to the $k$-element subsets of a set of $n$ elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. We show that for $ngeq 15k-24$, $K(n,k)$ is not semi-transitive, while for $kleq nleq 2k+1$, $K(n,k)$ is semi-transitive. Also, we show computationally that a subgraph $S$ on 16 vertices and 36 edges of $K(8,3)$, and thus $K(8,3)$ itself on 56 vertices and 280 edges, is non-semi-transitive. $S$ and $K(8,3)$ are the first explicit examples of triangle-free non-semi-transitive graphs, whose existence was established via ErdH{o}s theorem by Halld{o}rsson et al. in 2011. Moreover, we show that the complement graph $overline{K(n,k)}$ of $K(n,k)$ is semi-transitive if and only if $ngeq 2k$.
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