This thesis constructs special graphs with given circular chromatic numbers or circular flow numbers. Suppose $G=(V,E)$ is a graph and $rgeq 2$ is a real number. An $r$-coloring of a graph $G$ is a mapping $f:V ightarrow [0,r)$ such that for any adjacent vertices $x,y$ of $G$, $1leq |f(x)-f(y)|leq r-1$. The circular chromatic number $chi_c(G)$ is the least $r$ for which there exists an $r$-coloring of $G$. The circular chromatic number was introduced by Vince in 1988 in cite{vince}, where the parameter is called the {em star chromatic number} and denoted by $chi^*(G)$. Vince proved that for any rational number $k/dgeq 2$ there is a graph $G$ with $chi_c(G)=k/d$. In this thesis, we are interested in the existence of special graphs with given circular chromatic numbers. A graph $H$ is called a minor of a graph $G$ if $H$ can be obtained from $G$ by deleting some vertices and edges, and contracting some edges. A graph $G$ is called $H$-minor free if $H$ is not a minor of G. The well-known Hadwiger's conjecture asserts that for any positive integer $n$, any $K_n$-minor free graph $G$ is $(n-1)$-colorable. If this conjecture is true, then for any $K_n$-minor free graph $G$, we have $chi_c(G)leq n-1$. On the other hand, for any graph $G$ with at least one edge we have $chi_c(G)geq 2$. A natural question is this: Is it true that for any rational number $2leq rleq n-1$, there exist a $K_n$-minor free graph $G$ with $chi_c(G)=r$? For $n=4$, the answer is ``no". It was proved by Hell and Zhu in cite{hz98} that if $G$ is a $K_4$-minor free graph then either $chi_c(G)=3$ or $chi_c(G)leq 8/3$. So none of the rational numbers in the interval $(8/3,3)$ is the circular chromatic number of a $K_4$-minor free graph. For $ngeq 5$, Zhu cite{survey} proved that for any rational number $rin[2,n-2]$, there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$. The question whether there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$ for each rational number $rin(n-2,n-1)$ remained open. In this thesis, we answer this question in the affirmative. For each integer $ngeq 5$, for each rational number $rin[n-2,n-1]$, we construct a $K_n$-minor free graph $G$ with $chi_c(G)=r$. This implies that for each $ngeq 5$, for each rational number $rin[2,n-1]$, there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$. In case $n=5$, the $K_5$-minor free graphs constructed in this thesis are actually planar graphs. So our result implies that for each rational number $rin[2,4]$, there exists a planar graph $G$ with $chi_c(G)=r$. This result was first proved by Moser cite{moser} and Zhu cite{3-4}. To be precise, Moser cite{moser} proved that for each rational number $rin[2,3]$, there exist a planar graph $G$ with $chi_c(G)=r$, and Zhu cite{3-4} proved that for each rational number $rin[3,4]$, there exists a planar graph $G$ with $chi_c(G)=r$. Moser's and Zhu's proofs are quite complicated. Our construction is conceptually simpler. Moreover, for $ngeq 5$, $K_n$-minor free graphs, including the planar graphs are constructed with a unified method. For $K_4$-minor free graphs, although Hell and Zhu cite{hz98} proved that there is no $K_4$-minor free graph $G$ with $chi_c(G)in (8/3,3)$. The question whether there exists a $K_4$-minor free graph $G$ with $chi_c(G)=r$ for each rational number $rin[2,8/3]$ remained open. This thesis solves this problem: For each rational number $rin[2,8/3]$, we shall construct a $K_4$-minor free $G$ with $chi_c(G)=r$. This thesis also studies the relation between the circular chromatic number and the girth of $K_4$-minor free graphs. For each integer $n$, the supremum of the circular chromatic number of $K_4$-minor free graphs of odd girth (the length of shortest odd cycle) at least $n$ is determined. It is also proved that the same bound is sharp for $K_4$-minor free graphs of girth $n$. By a classical result of ErdH{o}s, for any positive integers $l$ and $n$, there exists a graph $G$ of girth at least $l$ and of chromatic number $n$. Using probabilistic method, Zhu cite{unique} proved that for each integer $l$ and each rational number $rgeq 2$, there is a graph $G$ of girth at least $l$ such that $chi_c(G)=r$. Construction of such graphs for $rgeq 3$ was given by Nev{s}etv{r}il and Zhu cite{nz}. The question of how to construct large girth graph $G$ with $chi_c(G)=r$ for given $rin(2,3)$ remained open. In this thesis, we present a unified method that constructs, for any $rgeq 2$, a graph $G$ of girth at least $l$ with circular chromatic number $chi_c(G) =r$. Graphs $G$ with $chi_c(G)=chi(G)$ have been studied extensively in the literature. Many families of graphs $G$ are known to satisfy $chi_c(G)=chi(G)$. However it remained as an open question as how to construct arbitrarily large $chi$-critical graphs $G$ of bounded maximum degree with $chi_c(G)=chi(G)$. This thesis presents a construction of such graphs. The circular flow number $Phi_c(G)$ is the dual concept of $chi_c(G)$. Let $G$ be a graph. Replace each edge $e=xy$ by a pair of opposite arcs $a=overrightarrow{xy}$ and $a^{-1}=overrightarrow{yx}$. We obtain a symmetric directed graph. Denote by $A(G)$ the set of all arcs of $G$. A chain is a mapping $f:A(G) ightarrow I!!R$ such that for each arc $a$, $f(a^{-1})=-f(a)$. A flow is a chain such that for each subset $X$ of $V(G)$, $sum_{ain[X,ar{X}]}f(a)=0$, where $[X,ar{X}]$ is the set of all arcs from $X$ to $V-X$. An $r$-flow is a flow such that for any arc $ain A(G)$ , $1leq |f(a)| leq r-1$. The circular flow number of $G$ is $Phi_c(G)=mbox{ inf}{r: G mbox{ admits a } rmbox{-flow}}$. It was conjectured by Tutte that every graph $G$ has $Phi_c(G)leq 5$. By taking the geometrical dual of planar graphs, Moser's and Zhu's results concerning circular chromatic numbers of planar graphs imply that for each rational number $rin[2,4]$, there is a graph $G$ with $Phi_c(G)=r$. The question remained open whether for each $rin(4,5)$, there exists a graph $G$ with $Phi_c(G)=r$. In this thesis, for each rational number $rin [4,5]$, we construct a graph $G$ with $Phi_c(G)=r$.

Sverige idag genererar byggbranschen 12 miljoner ton avfall per år. Enligt svensk lag ska avfallet sorteras för att öka möjligheten till återvinning och återbruk. Byggbranschen står för drygt 20 procent av växthusgasutsläppen i Sverige och genom att återbruka byggmaterial kan denna siffra reduceras. Avsikten med detta examensarbete är att lyfta ett antal möjligheter för att minska klimatavtrycket från byggbranschen. Föreliggande studie syftar därför till att fokusera på de framgångsfaktorer som identifierats hos utmärkande projekt på Skanska gällande återbruk av byggmaterial och produkter. Syftet med studien är att undersöka och belysa de utmaningar som finns med att återbruka samt att uppmärksamma tillvägagångssätt för hur arbete med att återbruka kan ökas.Studien har baserats på intervjuer samt litteratursökning inom ämnet återbruk. Fem deltagare i olika projekt hos Skanska har intervjuats där återbruk varit en del av processen. Arbetet har begränsats till fem intervjudeltagare då det ansågs vara tillräckligt för att besvara studiens syfte. Studien har avgränsats till återbrukat byggmaterial och produkter från byggnader, där material från anläggningsarbeten inte inkluderas. Arbetet skrivs i samarbete med Skanska.De framgångsfaktorer som har identifierats i resultatet är att det är betydelsefullt att lägga god tid på materialinventering och att systematiskt dokumentera de material som finns att tillgå, samt att planering av återbruk sker tidigt i processen. En annan framgångsfaktor för att lyckas med återbruk är att involvera externa aktörer som kan förmedla materialet vidare. De utmaningar som finns med att återbruka byggmaterial och produkter är de krav som ställs genom myndigheter och i lagar. Även kunskapsnivån är bristfällig när det kommer till återbruk i branschen och därmed problematiseras processen. För att anamma återbruk i framtiden kommer det cirkulära byggandet och det långsiktiga perspektivet vara av betydelse inom byggbranschen. Ett sätt för att arbeta med återbruk är att involvera projektörer tidigt i processen genom att förse dem med återbrukat material som går att projektera in i projektet. Vid projekteringen av byggnader bör även materialens framtida återbruk finnas i åtanke. När återbruk blir en del av fler byggprojekt kommer det medföra att avfallet minimeras, det kommer främja företagens hållbarhetsarbete samt att klimatpåverkan från branschen med största sannolikhet kommer minska. / The construction industry generates 12 million tons of waste annually in Sweden alone. The waste according to Swedish law has to be sorted in order to increase the possibility of recycling and reuse. The construction industry produces 20 percent of the Swedish greenhouse gas emissions and by reusing building materials this number can be reduced. The intention of this bachelor's thesis is to illustrate possibilities for reducing the industry's climate footprint. Following thesis intends to focus on identifying the success factors of prominent projects regarding reuse of building materials and products at Skanska. The purpose of the thesis is to explore and illuminate the challenges that come with reusing and detect approaches on how to increase work with reused materials.The thesis is based on interviews and literature in the subject of reuse. Five participants from different projects at Skanska where reuse was a part of the process was interviewed. The interviews were limited to five participants because it was considered to be enough to answer the purpose of the study. The study is limited to reuse of materials and products from buildings, waste from other parts of the industry is not included. This study is written in a collaboration with Skanska.The success factors that were found in this study is that it is important to assign the material listing phase a good amount of time, to systematically document the materials at hand and to plan to reuse early in the process. Another success factor to reusing is to involve external vendors that can help assign the material to new proprietors. The challenges that come with reusing building materials and products are the requirements set by authorities and laws. Also the level of knowledge in the industry about reuse is inadequate which complicates the process. The circular flow and long-time perspective will be of importance in order to appropriate reuse in the building industry in the future. One way to initiate this is by involving architects early in the process by providing them reused materials to plan into their project. In the design of new buildings, the future reuse of the materials should be kept in mind. When reuse becomes a part of more building projects it will result in less waste, the business’ sustainability work will advance and the climate footprint from the industry will most likely get reduced.