We study a very large family of graphs, the members of which comprise disjoint paths of
cliques ... more We study a very large family of graphs, the members of which comprise disjoint paths of
cliques with extremal cliques identified. This broad characterisation naturally generalises
those of various smaller families of graphs having well-known chromatic polynomials. We
derive a relatively simple formula for an arbitrary member of the subfamily consisting of
those graphs whose constituent clique-paths have at least one trivial extremal clique, and
use this formula to show that the set of all non-integer chromatic roots of these graphs is
closed under multiplication by natural numbers. A well-known result of Sokal then leads
to our main result, which is that there exists a set of chromatic roots which is closed under
positive integer multiplication in addition to being dense in the complex plane. Our findings
lend considerable weight to a conjecture of Cameron, who has suggested that this closure
property may be a generic feature of the chromatic polynomial. We also hope that the
formula we provide will be of use to those computing with chromatic polynomials.
We define a biclique to be the complement of a bipartite graph, consisting
of two cliques joined... more We define a biclique to be the complement of a bipartite graph, consisting
of two cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the chromatic
polynomial of an arbitrary biclique, and use this to give certain conditions under which
two of the graphs have chromatic polynomials with the same splitting field. Finally,
we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by
showing that for any cubic integer α, there is a natural number n such that α + n is a
chromatic root.
The multivariate Tutte polynomial Z_M of a matroid M is a generalization
of the standard two-var... more The multivariate Tutte polynomial Z_M of a matroid M is a generalization
of the standard two-variable version, obtained by assigning a separate variable v_e
to each element e of the ground set E. It encodes the full structure of M. Let v =
{v_e}e∈E, let K be an arbitrary field, and suppose M is connected. We show that Z_M
is irreducible over K(v), and give three self-contained proofs that the Galois group
of Z_M over K(v) is the symmetric group of degree n, where n is the rank of M.
An immediate consequence of this result is that the Galois group of the multivariate
Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we
conjecture a similar result for the standard Tutte polynomial of a connected matroid.
We study a very large family of graphs, the members of which comprise disjoint paths of
cliques ... more We study a very large family of graphs, the members of which comprise disjoint paths of
cliques with extremal cliques identified. This broad characterisation naturally generalises
those of various smaller families of graphs having well-known chromatic polynomials. We
derive a relatively simple formula for an arbitrary member of the subfamily consisting of
those graphs whose constituent clique-paths have at least one trivial extremal clique, and
use this formula to show that the set of all non-integer chromatic roots of these graphs is
closed under multiplication by natural numbers. A well-known result of Sokal then leads
to our main result, which is that there exists a set of chromatic roots which is closed under
positive integer multiplication in addition to being dense in the complex plane. Our findings
lend considerable weight to a conjecture of Cameron, who has suggested that this closure
property may be a generic feature of the chromatic polynomial. We also hope that the
formula we provide will be of use to those computing with chromatic polynomials.
We define a biclique to be the complement of a bipartite graph, consisting
of two cliques joined... more We define a biclique to be the complement of a bipartite graph, consisting
of two cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the chromatic
polynomial of an arbitrary biclique, and use this to give certain conditions under which
two of the graphs have chromatic polynomials with the same splitting field. Finally,
we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by
showing that for any cubic integer α, there is a natural number n such that α + n is a
chromatic root.
The multivariate Tutte polynomial Z_M of a matroid M is a generalization
of the standard two-var... more The multivariate Tutte polynomial Z_M of a matroid M is a generalization
of the standard two-variable version, obtained by assigning a separate variable v_e
to each element e of the ground set E. It encodes the full structure of M. Let v =
{v_e}e∈E, let K be an arbitrary field, and suppose M is connected. We show that Z_M
is irreducible over K(v), and give three self-contained proofs that the Galois group
of Z_M over K(v) is the symmetric group of degree n, where n is the rank of M.
An immediate consequence of this result is that the Galois group of the multivariate
Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we
conjecture a similar result for the standard Tutte polynomial of a connected matroid.
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cliques with extremal cliques identified. This broad characterisation naturally generalises
those of various smaller families of graphs having well-known chromatic polynomials. We
derive a relatively simple formula for an arbitrary member of the subfamily consisting of
those graphs whose constituent clique-paths have at least one trivial extremal clique, and
use this formula to show that the set of all non-integer chromatic roots of these graphs is
closed under multiplication by natural numbers. A well-known result of Sokal then leads
to our main result, which is that there exists a set of chromatic roots which is closed under
positive integer multiplication in addition to being dense in the complex plane. Our findings
lend considerable weight to a conjecture of Cameron, who has suggested that this closure
property may be a generic feature of the chromatic polynomial. We also hope that the
formula we provide will be of use to those computing with chromatic polynomials.
of two cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the chromatic
polynomial of an arbitrary biclique, and use this to give certain conditions under which
two of the graphs have chromatic polynomials with the same splitting field. Finally,
we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by
showing that for any cubic integer α, there is a natural number n such that α + n is a
chromatic root.
of the standard two-variable version, obtained by assigning a separate variable v_e
to each element e of the ground set E. It encodes the full structure of M. Let v =
{v_e}e∈E, let K be an arbitrary field, and suppose M is connected. We show that Z_M
is irreducible over K(v), and give three self-contained proofs that the Galois group
of Z_M over K(v) is the symmetric group of degree n, where n is the rank of M.
An immediate consequence of this result is that the Galois group of the multivariate
Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we
conjecture a similar result for the standard Tutte polynomial of a connected matroid.
cliques with extremal cliques identified. This broad characterisation naturally generalises
those of various smaller families of graphs having well-known chromatic polynomials. We
derive a relatively simple formula for an arbitrary member of the subfamily consisting of
those graphs whose constituent clique-paths have at least one trivial extremal clique, and
use this formula to show that the set of all non-integer chromatic roots of these graphs is
closed under multiplication by natural numbers. A well-known result of Sokal then leads
to our main result, which is that there exists a set of chromatic roots which is closed under
positive integer multiplication in addition to being dense in the complex plane. Our findings
lend considerable weight to a conjecture of Cameron, who has suggested that this closure
property may be a generic feature of the chromatic polynomial. We also hope that the
formula we provide will be of use to those computing with chromatic polynomials.
of two cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the chromatic
polynomial of an arbitrary biclique, and use this to give certain conditions under which
two of the graphs have chromatic polynomials with the same splitting field. Finally,
we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by
showing that for any cubic integer α, there is a natural number n such that α + n is a
chromatic root.
of the standard two-variable version, obtained by assigning a separate variable v_e
to each element e of the ground set E. It encodes the full structure of M. Let v =
{v_e}e∈E, let K be an arbitrary field, and suppose M is connected. We show that Z_M
is irreducible over K(v), and give three self-contained proofs that the Galois group
of Z_M over K(v) is the symmetric group of degree n, where n is the rank of M.
An immediate consequence of this result is that the Galois group of the multivariate
Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we
conjecture a similar result for the standard Tutte polynomial of a connected matroid.