46 research outputs found
Explicit construction of Ramanujan bigraphs
We construct explicitly an infinite family of Ramanujan graphs which are
bipartite and biregular. Our construction starts with the Bruhat-Tits building
of an inner form of . To make the graphs finite, we take
successive quotients by infinitely many discrete co-compact subgroups of
decreasing size.Comment: 10 page
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
Minimum Cost Homomorphisms to Reflexive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)H, the {\em minimum cost homomorphism problem} for HHGc_i(u)u\in V(G)i\in V(H)kGHk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs HHHH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <i<j, s<rir, js
\in A(H)is \in A(H)jr \in A(H)H$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs
Left-cut-percolation and induced-Sidorenko bigraphs
A Sidorenko bigraph is one whose density in a bigraphon is minimized
precisely when is constant. Several techniques of the literature to prove
the Sidorenko property consist of decomposing (typically in a tree
decomposition) the bigraph into smaller building blocks with stronger
properties. One prominent such technique is that of -decompositions of
Conlon--Lee, which uses weakly H\"{o}lder (or weakly norming) bigraphs as
building blocks. In turn, to obtain weakly H\"{o}lder bigraphs, it is typical
to use the chain of implications reflection bigraph cut-percolating
bigraph weakly H\"{o}lder bigraph. In an earlier result by the
author with Razborov, we provided a generalization of -decompositions,
called reflective tree decompositions, that uses much weaker building blocks,
called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs.
In this paper, we show that "left-sided" versions of the concepts of
reflection bigraph and cut-percolating bigraph yield a similar chain of
implications: left-reflection bigraph left-cut-percolating bigraph
induced-Sidorenko bigraph. We also show that under mild hypotheses,
the "left-sided" analogue of the weakly H\"{o}lder property (which is also
obtained via a similar chain of implications) can be used to improve bounds on
another result of Conlon--Lee that roughly says that bigraphs with enough
vertices on the right side of each realized degree have the Sidorenko property.Comment: 42 pages, 5 figure