On set intersection representations of graphs

Abstract

The intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets A x ⊆ {1, . . . ,t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only ifThe weight of such a representation is the sum ∑ x |A x | over all vertices x. We exhibit explicit bipartite n × n graphs whose intersection dimension is: (i) at least n 1/|L| with respect to any type L, (ii) at least √ n with respect to any type of the form L = {k, k + 1, . . .}, and (iii) at least n 1/|R| with respect to any type of the form L = {k | k mod p ∈ R}, where p is a prime number. We also show that any intersection representation of a Hadamard graph must have weight about n ln n/ ln ln n, independent on the used type L. Finally, we formulate several problems about intersection dimensions of graphs related to some basic open problems in the complexity of boolean functions.

On the Fine-Grained Complexity of Rainbow Coloring

“…However, one can observe that for any biclique (a complete bipartite subgraph) in the graph (V, V 2 \ S) it is sufficient to use just one such vertex x (connected to all the vertices of the biclique). By applying a result of Jukna [16] we can show that in our specific instance of Subset Rainbow 2-Coloring which results from a 3-SAT formula, the number of bicliques needed to cover all the pairs in V 2 \ S is small enough. We show a 2 |V (G)| |V (G)| O(1) -time algorithm to find such a cover.…”

“…FindColoring(S 0 , c 0 , f ) 1 if S0 = ∅ then 2 return c03 if for some r ∈ S0 there are edges e1, e2 ∈ f (r) with c0(e1) = c0(e2) then Pick any {u, v} ∈ S0;6 Find any f -guided u-v walk W of length at most k using Lemma 19; 7 if W does not exist then Let c1 be obtained from c0 by coloring the uncolored edges of W to get a rainbow walk;10 if FindColoring(S0 \ {u, v}, c1, f | S 0 \{u,v} ) = null then return the coloring found;11 for e ∈ E(W ) \ Dom(c0) do 12 for α ∈ [k] do 13 for r ∈ S0 \ {{u, v}} do 14Let ce,α be obtained from c0 by coloring e with α;15 Let fe,r be obtained from f by putting f (r) := f (r) ∪ {e};16 if FindColoring(S0, ce,α, fe,r) = null then return the coloring found;17 return null some of the uncolored edges e of W into c 1 (e) (instead of some color α) makes some other request r ∈ S 0 \ {{u, v}} impossible to satisfy. For every possible triple (e, α, r) we invoke FindColoring with the same set of requests S 0 , partial coloring c 0 extended by coloring e with α, and the guide function f extended by putting f (r) := f (r) ∪ {e}.…”

Biclique Covers and Partitions

“…An intersection graph of a family of subsets of some groundset has one vertex for each of the sets, and an edge between two vertices if and only if the corresponding sets intersect. See [10] or [15] for more on intersection graphs for various set families. In this paper, we are interested in intersection graphs where all sets in the family are subcubes of a hypercube.…”

Secret-Sharing Schemes for Very Dense Graphs

“…We next quote a lemma of Jukna [34] on the existence of small bipartite covers. For completeness we present the proof of this lemma.…”

“…There are not too many such vertices in the complement graph, and the share size in realizing each star (namely, a vertex and its adjacent edges) is at most n. Once we removed all edges adjacent to vertices whose degree is "big", we use the cover by equivalence graphs to cover the remaining edges. To achieve a better scheme, we first remove vertices of high degree using stars, then use covers of bipartite graphs of [34] to further reduce the degree of the vertices in the complement graph, and finally use the cover by equivalence graphs.…”