Fulkerson′s Conjecture and Circuit Covers

SCHUR COVERS and CARLITZ’S CONJECTURE

We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomial f over a finite field Fq is a polynomial that is a permutation polynomial on infinitely many finite extensions of Fq. Carlitz’s conjecture says f must be of odd degree (if q is odd). Indeed, excluding characteristic 2 and 3, arithmetic ...

Circuit and fractional circuit covers of matroids

LetM be a connectedmatroid having a ground set E. Lemos andOxley proved that |E(M)| ≤ 2 c(M)c (M)where c(M) (resp. c(M)) is the circumference (resp. cocircumference) of M. In addition, they conjectured that one can find a collection of at most c(M) circuits which cover the elements ofM at least twice. In this paper, we verify this conjecture for regular matroids. Moreover, we show that a versio...

On semiextensions and circuit double covers

We introduce a concept of a semiextension of a cycle, and we conjecture a simple necessary and sufficient condition for its existence. It is shown that our conjecture implies a strong form of the circuit double cover conjecture. We prove that the conjecture is equivalent to its restriction to cubic graphs, and we show that it holds for every cycle which is a spanning subgraph of the given graph...

Kotzig frames and circuit double covers

A cubic graph H is called a Kotzig graph if H has a circuit double cover consisting of three Hamilton circuits. It was first proved by Goddyn that if a cubic graph G contains a spanning subgraph H which is a subdivision of a Kotzig graph then G has a circuit double cover. A spanning subgraph H of a cubic graph G is called a Kotzig frame if the contracted graph G/H is even and every non-circuit ...

Circuit Covers in Series-Parallel Mixed Graphs