35,422 research outputs found
The DP Color Function of Clique-Gluings of Graphs
DP-coloring (also called correspondence coloring) is a generalization of list
coloring that has been widely studied in recent years after its introduction by
Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial
of a graph , , the DP color function of , denoted ,
counts the minimum number of DP-colorings over all possible -fold covers.
Formulas for chromatic polynomials of clique-gluings of graphs are well-known,
but the effect of such gluings on the DP color function is not well understood.
In this paper we study the DP color function of -gluings of graphs.
Recently, Becker et. al. asked whether whenever , where the expression on the right is the DP-coloring analogue of the
corresponding chromatic polynomial formula for a -gluing of . Becker et. al. showed this inequality holds when . In this paper we
show this inequality holds for edge-gluings (). On the other hand, we show
it does not hold for triangle-gluings (), which also answers a question of
Dong and Yang (2021). Finally, we show a relaxed version, based on a class of
-fold covers that we conjecture would yield the fewest DP-colorings for a
given graph, of the inequality holds when .Comment: 20 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:2104.1226
Pseudo-Codewords of Cycle Codes via Zeta Functions
Cycle codes are a special case of low-density parity-check (LDPC) codes and
as such can be decoded using an iterative message-passing decoding algorithm on
the associated Tanner graph. The existence of pseudo-codewords is known to
cause the decoding algorithm to fail in certain instances. In this paper, we
draw a connection between pseudo-codewords of cycle codes and the so-called
edge zeta function of the associated normal graph and show how the Newton
polyhedron of the zeta function equals the fundamental cone of the code, which
plays a crucial role in characterizing the performance of iterative decoding
algorithms.Comment: Presented at Information Theory Workshop (ITW), San Antonio, TX, 200
Covers of Point-Hyperplane Graphs
We construct a cover of the non-incident point-hyperplane graph of projective
dimension 3 for fields of characteristic 2. If the cardinality of the field is
larger than 2, we obtain an elementary construction of the non-split extension
of SL_4 (F) by F^6.Comment: 10 pages, 3 figure
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