Xiangqian Zhou

Generating an internally 4-connected binary matroid from another

Abstract We prove that if N is an internally 4-connected proper minor of a weakly 4-connected binary matroid M with | E ( N ) | ≥ 7 , then either there exists a weakly 4-connected minor M ′ of M such that M ′ has an N -minor and 1 ≤ | E ( M ) | − | E ( M ′ ) | ≤ 2 , or one of M and M ∗ is isomorphic to D n , D n ∖ f 1 , D n , or D n ∖ f 1 .

Note: On clone sets of GF( q)-representable matroids

We bound the size of a clone set in a 3-connected non-uniform GF(q)-representable matroid by a linear function of q. This bound is given by investigating the representability of a class of near-uniform matroids.

The strong chromatic index of (3,)-bipartite graphs

A strong edge-coloring of a graph G=(V,E) is a partition of its edge set E into induced matchings. We study bipartite graphs with one part having maximum degree at most 3 and the other part having maximum degree . We show that every such graph has a strong edge-coloring using at most 3 colors. Our result confirms a conjecture of Brualdi and Quinn Massey (1993) for this class of bipartite graphs.

Bicircular matroids representable over GF(4) or GF(5)

Given a bicircular matroid B ( G ) and q ź { 4 , 5 } , we characterize when the bicircular matroid B ( G ) is G F ( q ) -representable by precisely describing the structure of G . These descriptions yield polynomial-time algorithms with input G to certify if B ( G ) is or is not G F ( q ) -representable.

Clonal sets in GF(q)-representable matroids

Whittle [12] conjectured that if MM is a 33-connected quaternary matroid with a clonal pair {e,f}{e,f}, then M∖e,fM∖e,f and M/e,fM/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9}q∈{4,5,7,8,9} if MM is a 3-connected GF(q)GF(q)-representable matroid with a clonal set XX of size q−2q−2, then M∖XM∖X and M/XM/X are binary.

Clones in matroids representable over a prime field

Abstract We show that for every prime number p , a 3-connected non-uniform G F ( p ) -representable matroid can have a clone set of size at most p − 2 .

Clones in 3-connected frame matroids

We determine the structure of clonal classes of 3-connected frame matroids in terms of the structure of biased graphs. Robbins has conjectured that a 3-connected non-uniform matroid with a clonal class of size q - 1 is not G F ( q ) -representable. We confirm the conjecture for the class of frame matroids.

Bicircular matroids representable over G F ( 4 ) or G F ( 5 )

Given a bicircular matroid B ( G ) and q ź { 4 , 5 } , we characterize when the bicircular matroid B ( G ) is G F ( q ) -representable by precisely describing the structure of G . These descriptions yield polynomial-time algorithms with input G to certify if B ( G ) is or is not G F ( q ) -representable.

Bicircular matroids representable over GF(4)GF(4) or GF(5)GF(5)

Given a bicircular matroid B ( G ) and q ź { 4 , 5 } , we characterize when the bicircular matroid B ( G ) is G F ( q ) -representable by precisely describing the structure of G . These descriptions yield polynomial-time algorithms with input G to certify if B ( G ) is or is not G F ( q ) -representable.

Some minor-closed classes of signed graphs

Abstract We define four minor-closed classes of signed graphs in terms of embeddability in the annulus, projective plane, torus, and Klein bottle. We give the full list of 20 excluded minors for the smallest class and make a conjecture about the largest class.