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

Short directed cycles in bipartite digraphs

The Caccetta-H\"aggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $\alpha, \beta> 0$ with $k\alpha +\beta>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $\beta|B|$, and every vertex in $B$ has out-degree at least $\alpha|A|$, then $G$ has a directed cycle of length at most $2k$. This implies the Caccetta-H\"aggkvist conjecture (set $\beta>0$ and very small), and again is best possible for infinitely many pairs $(\alpha,\beta)$. We prove this for $k = 1,2$, and prove a weaker statement (that $\alpha+\beta>2/(k+1)$ suffices) for $k=3,4$

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Gy\'arf\'as and Sumner independently conjectured that for every tree $T$, the class of graphs not containing $T$ as an induced subgraph is $\chi$-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 $T$, but has been proved for some particular trees. For $k\ge 1$, let us say a broom of length $k$ is a tree obtained from a $k$-edge path with ends $a,b$ by adding some number of leaves adjacent to $b$, and we call $a$ its handle. A tree obtained from brooms of lengths $k_1,...,k_n$ by identifying their handles is a $(k_1,...,k_n)$-multibroom. Kierstead and Penrice proved that every $(1,...,1)$-multibroom $T$ satisfies the Gy\'arf\'as-Sumner conjecture, and Kierstead and Zhu proved the same for $(2,...,2)$-multibrooms. In this paper give a common generalization: we prove that every $(1,...,1,2,...,2)$-multibroom satisfies the Gy\'arf\'as-Sumner conjecture

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 ‘Table A providers’ in Australian higher education and enrolments in seven equity groups: Low Socio-Economic Status (‘low SES’) 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 ‘Culturally and Linguistically Diverse’ or ‘CALD’ 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) – 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 – CSU, Southern Cross, UNE, Federation, CQU, JCU, USQ, Tasmania, CDU, and Batchelor. Metropolitan Institutions with Regional Campuses: Institutions with one or more regional campus – 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)

Detecting an induced net subdivision

A {\em net} is a graph consisting of a triangle $C$ and three more vertices, each of degree one and with its neighbour in $C$, and all adjacent to different vertices of $C$. 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

The Minimal Automorphism-Free Tree

A finite tree $T$ with $|V(T)| \geq 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1 \preceq T_2$ if $T_1$ can be obtained from $T_2$ by successively deleting one leaf at a time in such a way that each intermediate tree is also automorphism-free. In this paper, we prove that $\mathcal{AFT}$ has a unique minimal element. This result gives an affirmative answer to the question asked by Rupinski.Comment: 8 pages, 5 figure