1,738 research outputs found
Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes
A hole in a graph is an induced subgraph which is a cycle of length at least
four. We prove that for every positive integer k, every triangle-free graph
with sufficiently large chromatic number contains holes of k consecutive
lengths
Induced subgraphs of graphs with large chromatic number. XIII. New brooms
Gy\'arf\'as and Sumner independently conjectured that for every tree , the
class of graphs not containing as an induced subgraph is -bounded,
that is, the chromatic numbers of graphs in this class are bounded above by a
function of their clique numbers. This remains open for general trees , but
has been proved for some particular trees. For , let us say a broom of
length is a tree obtained from a -edge path with ends by adding
some number of leaves adjacent to , and we call its handle. A tree
obtained from brooms of lengths by identifying their handles is a
-multibroom. Kierstead and Penrice proved that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture, and
Kierstead and Zhu proved the same for -multibrooms. In this paper
give a common generalization: we prove that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture
Short directed cycles in bipartite digraphs
The Caccetta-H\"aggkvist conjecture implies that for every integer ,
if is a bipartite digraph, with vertices in each part, and every vertex
has out-degree more than , then has a directed cycle of length at
most . If true this is best possible, and we prove this for
and all .
More generally, we conjecture that for every integer , and every pair
of reals with , if is a bipartite
digraph with bipartition , where every vertex in has out-degree at
least , and every vertex in has out-degree at least ,
then has a directed cycle of length at most . This implies the
Caccetta-H\"aggkvist conjecture (set and very small), and again is
best possible for infinitely many pairs . We prove this for , and prove a weaker statement (that suffices) for
Student equity performance in Australian higher education: 2007 to 2014
This NCSEHE Briefing Note provides an update on domestic undergraduate student enrolment and equity outcomes from 2007 to 2014, following Koshy and Seymour (2014). It focuses on undergraduate outcomes for Table A providers, given policy changes in recent years to Australian undergraduate education that affect them, including the full deregulation of undergraduate places in 2012 under the Demand Driven System (DDS). It reports on the number of domestic undergraduates between 2007 and 2014 in the 38 āTable A providersā in Australian higher education and enrolments in seven equity groups:
Low Socio-Economic Status (ālow SESā) students;
Students with Disability;
Indigenous Students;
Women in Non-Traditional Areas;
Regional Students;
Remote Students; and
Non-English Speaking Background (NESB) students (also referred to as āCulturally and Linguistically Diverseā or āCALDā students).
In each equity group, results are reported for the national system in total, by institutional groupings, by state and territory, and by regional or metropolitan status, for each year. All reporting is for domestic undergraduates in each given year. The institutional groupings in 2014 were as follows:
The Group of Eight: Australian National University (ANU), Melbourne, Monash, Sydney, New South Wales (UNSW), Queensland (UQ), Western Australia (UWA), and Adelaide.
The Australian Technology Network (ATN): Curtin University, University of Technology, Sydney (UTS), RMIT University (RMIT), Queensland University of Technology (QUT), and University of South Australia (UniSA).
The Innovative Research Universities (IRU): Murdoch, Flinders, Griffith, James Cook (JCU), La Trobe, Charles Darwin University (CDU) and Newcastle. (Note: Newcastle left the IRU in December 2014).
Regional Universities Network: Southern Cross, New England (UNE), Federation, Sunshine Coast (SCU), CQUniversity Australia (CQU), and Southern Queensland (USQ).
The Unaligned Universities: Other Table A providers) ā Macquarie, Wollongong, Deakin, Charles Sturt (CSU), Tasmania, Australian Catholic University (ACU), Canberra, Edith Cowan University (ECU), Swinburne, Victoria, Western Sydney (WSU) and The Batchelor Institute (Batchelor)
(Note: Batchelor and CDU entered into a collaborative partnership in 2012 which has seen CDU take delivery of most undergraduate programs.)
In addition, an analysis is reported for universities on the basis of their campus location and infrastructure, as per Koshy and Phillimore (2013):
Regionally Headquartered: Institutions with a major regional ā CSU, Southern Cross, UNE, Federation, CQU, JCU, USQ, Tasmania, CDU, and Batchelor.
Metropolitan Institutions with Regional Campuses: Institutions with one or more regional campus ā Newcastle, Sydney, Wollongong, Deakin, La Trobe, Monash, RMIT, Melbourne, QUT, UQ, SCU, Curtin, ECU, Murdoch, UWA, Flinders, Adelaide, UniSA, and ACU.
No Regional Campuses: Metropolitan Institutions with no regional campus: ANU, Sydney, UNSW, Griffith, Macquarie, Canberra, Swinburne, Victoria and WSU.
All student data reported or derived for the purposes of this document are sourced from Students: Selected Higher Education Statistics 2014 (Appendix 2: Equity Data), published by the Australian Government Department of Education and Training (2015)
Induced subgraphs of graphs with large chromatic number. XII. Distant stars
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph
with bounded clique number and very large chromatic number contains H as an
induced subgraph. This is still open, although it has been proved for a few
simple families of trees, including trees of radius two, some special trees of
radius three, and subdivided stars. These trees all have the property that
their vertices of degree more than two are clustered quite closely together. In
this paper, we prove the conjecture for two families of trees which do not have
this restriction. As special cases, these families contain all double-ended
brooms and two-legged caterpillars
Detecting a long odd hole
For each integer , we give a polynomial-time algorithm to test
whether a graph contains an induced cycle with length at least and odd
Detecting an induced net subdivision
A {\em net} is a graph consisting of a triangle and three more vertices,
each of degree one and with its neighbour in , and all adjacent to different
vertices of . We give a polynomial-time algorithm to test whether an input
graph has an induced subgraph which is a subdivision of a net. Unlike many
similar questions, this does not seem to be solvable by an application of the
"three-in-a-tree" subroutine
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