Short Cycle Covers of Cubic Graphs and Graphs with Minimum Degree Three

The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx 1.630m$

Signed circuit 66-covers of signed K4K_4-minor-free graphs

Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit $k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is the well-know circuit double cover conjecture. For signed circuit $k$-cover of signed graphs, it is known that for every integer $k\leq 5$, there are infinitely many coverable signed graphs without signed circuit $k$-cover and there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't admit a signed circuit $1$-cover. Fan conjectured that every coverable signed graph has a signed circuit $6$-cover. This conjecture was verified only for signed eulerian graphs and for signed graphs whose bridgeless-blocks are eulerian. In this paper, we prove that this conjecture holds for signed $K_4$-minor-free graphs. The $6$-cover is best possible for signed $K_4$-minor-free graphs

Integer Flows and Circuit Covers of Graphs and Signed Graphs

The work in Chapter 2 is motivated by Tutte and Jaeger\u27s pioneering work on converting modulo flows into integer-valued flows for ordinary graphs. For a signed graphs (G, sigma), we first prove that for each k ∈ {lcub}2, 3{rcub}, if (G, sigma) is (k -- 1)-edge-connected and contains an even number of negative edges when k = 2, then every modulo k-flow of (G, sigma) can be converted into an integer-valued ( k + 1)-ow with a larger or the same support. We also prove that if (G, sigma) is odd-(2p+1)-edge-connected, then (G, sigma) admits a modulo circular (2 + 1/ p)-flows if and only if it admits an integer-valued circular (2 + 1/p)-flows, which improves all previous result by Xu and Zhang (DM2005), Schubert and Steffen (EJC2015), and Zhu (JCTB2015).;Shortest circuit cover conjecture is one of the major open problems in graph theory. It states that every bridgeless graph G contains a set of circuits F such that each edge is contained in at least one member of F and the length of F is at most 7/5∥E(G)∥. This concept was recently generalized to signed graphs by Macajova et al. (JGT2015). In Chapter 3, we improve their upper bound from 11∥E( G)∥ to 14/3 ∥E(G)∥, and if G is 2-edgeconnected and has even negativeness, then it can be further reduced to 11/3 ∥E(G)∥.;Tutte\u27s 3-flow conjecture has been studied by many graph theorists in the last several decades. As a new approach to this conjecture, DeVos and Thomassen considered the vectors as ow values and found that there is a close relation between vector S1-flows and integer 3-NZFs. Motivated by their observation, in Chapter 4, we prove that if a graph G admits a vector S1-flow with rank at most two, then G admits an integer 3-NZF.;The concept of even factors is highly related to the famous Four Color Theorem. We conclude this dissertation in Chapter 5 with an improvement of a recent result by Chen and Fan (JCTB2016) on the upperbound of even factors. We show that if a graph G contains an even factor, then it contains an even factor H with.;∥E(H)∥ ≥ 4/7 (∥ E(G)∥+1)+ 1/7 ∥V2 (G)∥, where V2( G) is the set of vertices of degree two

Disparities in the distribution of municipal services

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Civil Engineering; and, (M.C.P.)--Massachusetts Institute of Technology, Dept. of Urban Studies and Planning, 1971.Bibliography: leaves 148-151.by Robert Marlay.M.C.P.M.S

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Class Notes in Discrete Mathematics, Operations Research, Statistics and Probability (Fourth Edition, v1)

Editorâ⠡‰ ¾Ã‚¢s Note: In graduate school, it became too cumbersome for me to look-up equations, theorems, proofs, and problem solutions from previous courses. I had three boxes full of notes and was going on my fourth. Due to the need to reference my notes periodically, the notes became more unorganized over time. Thatâ⠡‰ ¾Ã‚¢s when I decided to typeset them. I have been doing this for over a decade. Later in life, some colleagues asked if I could make these notes available to others (they were talking about themselves). I did. These notes can be downloaded for free from the web site http://www.repec.org/ and can be found in the Library of Congress. Note that the beginning of each chapter lists the professorâ⠡‰ ¾Ã‚¢s name and aï⠡½liation. Additionally, the course number, the date the course was taken, and the text book are given. The reader may also notice that I have made more use of the page space than in the previous editions of this manuscript. Hence, the book is shorter. If this causes the reader problems, then simply copy the proofs onto a blank sheet of paper â⠡¬â€� one line per algebraic manipulation. In this text, I put several algebraic manipulations on one line to save space