An attempt to classify bipartite graphs by their chromatic Polynomial.

For the purpose of tackling the four-colour problem, Birkhoff (1912) introduced the chromatic polynomial of a map, denoted by P(M,A), which is a number of proper Acolouring of a map M. Whitney (1932), who established many fundamental results for it, later generalized the notion of a chromatic polynomial to that of an arbitrary graph. In 1968, Read asked whether it is possible to find a set of necessary and sufficient algebraic conditions for a polynomial to be the chromatic polynomial of some graph. In particular, Read asked for a necessary and sufficient condition for two graphs to be chromatically equivalent; that is, to have the same chromatic polynomial. In 1978, Chao and Whitehead defined a graph to be chromatically unique if no other graphs share its chromatic polynomial. Since then many researchers have been studying chromatic uniqueness and chromatic equivalence of graphs

Chromaticity of Certain Bipartite Graphs

Since the introduction of the concepts of chromatically unique graphs and chromatically equivalent graphs, numerous families of such graphs have been obtained. The purpose of this thesis is to continue with the search of families of chromatically unique bipartite graphs. In Chapters 1 and 2, we define the concept of graph colouring, the associated chromatic polynomial and some properties of a chromatic polynomial. We also give some necessary conditions for graphs that are chromatically unique or chromatically equivalent. We end this chapter by stating some known results on the chromaticity of bipartite graphs, denoted as K(p,q)

Chromaticity of a family of 5-partite graphs

AbstractLet P(G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G∼H, if P(G,λ)=P(H,λ). We write [G]={H∣H∼G}. If [G]={G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n vertices according to the number of 6-independent partitions of G. Using these results, we investigate the chromaticity of G with certain stars or matching deleted parts . As a by-product, two new families of chromatically unique complete 5-partite graphs G with certain stars or matching deleted parts are obtained

Chromatically unique bipartite graphs with certain 3-independent partition numbers III

For integers p, q, s with p ≥ q ≥ 2 and s ≥ 0 , let ( ) 2 , K−s p q denote the set of 2_connected bipartite graphs which can be obtained from K(p,q) by deleting a set of s edges. In this paper, we prove that for any graph ( ) 2 G∈K−s p,q with p ≥ q ≥ 3 and 1 ≤ s ≤ q - 1 if the number of 3-independent partitions of G is 2p-1 + 2q-1 + s + 4, then G is chromatically unique. This result extends both a theorem by Dong et al.[2]; and results in [4] and [5]

ON GARLANDS IN x-UNIQUELY COLORABLE GRAPHS

a graph G is called x-uniquely colorable, if all its x-colorings induce the same partion of the vertex set into one-color components. For x-uniquely colorable graphs new bound of the number of vertex set partions into x + 1 cocliques is found. © 2019 gein p.a

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

Colorings of graphs, digraphs, and hypergraphs

Brooks' Theorem ist eines der bekanntesten Resultate über Graphenfärbungen: Sei G ein zusammenhängender Graph mit Maximalgrad d. Ist G kein vollständiger Graph, so lassen sich die Ecken von G so mit d Farben färben, dass zwei benachbarte Ecken unterschiedlich gefärbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem für Färbungen von Hypergraphen und gerichteten Graphen. Eine Färbung eines Hypergraphen ist eine Färbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. Kernstück ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,…,f_p) von Funktionen, welche von V(H) in die natürlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,…,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthält, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+…+f_p(v) \geq d_H(v) für alle v fast immer ausreichend für die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die Ausnahmefälle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-Färbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht Färbungen gerichteter Graphen. Eine azyklische Färbung eines gerichteten Graphen ist eine Färbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische Färbungsresultate übertragen. Dazu zählt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-Färbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-Färbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen für diese angegeben und die gerichtete Version des Satzes von Hajós bewiesen.Brooks‘ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,…,f_p) of functions, we examine if there is a partition of $H$ into induced subhypergraphs H_1,…,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v

Correspondence Colouring and its Applications to List Colouring and Delay Colouring

In this thesis, we study correspondence colouring and its applications to list colouring and delay colouring. We give a detailed exposition of the paper of Dvořák, and Postle introducing correspondence colouring. Moreover, we generalize two important results in delay colouring. The first is a result by Georgakopoulos, stating that cubic graphs are 4-delay colourable. We show that delay colouring can be formulated as an instance of correspondence colouring. Then we show that the modified line graph of a cubic bipartite graph is generally 4-correspondence colourable, using a Brooks’ type theorem for correspondence colouring. This allows us to give a more simple proof of a stronger result. The second result is one by Edwards and Kennedy, which states that quartic bipartite graphs are 5-delay colourable. We introduce the notion of p-cyclic correspondence colouring which is a type of correspondence colouring that generalizes delay colouring. We then prove that the modified line graph of a quartic bipartite graph is 5-cyclic correspondence colourable using the Combinatorial Nullstellensatz. We also show that the maximum DP-chromatic number of any cycle plus triangles (CPT) graph is 4. We construct a CPT graph with DP-chromatic number at least 4. Moreover, the upper bound follows easily from the Brooks’ type theorem for correspondence colouring. Finally, we do a preliminary investigation into using parity techniques in correspondence colouring to prove that CPT graphs are 3-choosable

Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm

Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's $\omega, \Delta, \chi$ Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex $v$ depends on its degree, denoted $d(v)$, and the size of the largest clique containing it, denoted $\omega(v)$. We make extensive use of the probabilistic method in this part. We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is $L$-colorable if $L$ is a list-assignment such that $$|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil$$ for each vertex $v$ and $\varepsilon \leq 1/2$, and we prove this for $\varepsilon \leq 1/330$ under some mild additional assumptions. We also conjecture the ``$\mathrm{mad}$ version'' of Reed's Conjecture, even for list-coloring. That is, for $\varepsilon \leq 1/2$, every graph $G$ satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where $\mathrm{mad}(G)$ is the maximum average degree of $G$. We prove this conjecture for small values of $\varepsilon$, assuming $\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$. We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of $K_t$-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for $\Delta$ sufficiently large, if $G$ is a graph of maximum degree at most $\Delta$ with list-assignment $L$ such that for each vertex $v\in V(G)$, $$|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\}$$ and $d(v) \geq \ln^2\Delta$, then $G$ is $L$-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph $G$ with $\omega(G)\leq \omega$ and $\Delta(G)\leq \Delta$ for $\Delta$ sufficiently large: $$\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}.$$ In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the $[0, 1]$-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the $\omega, \Delta, \chi$ paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if $G$ is a graph and $f : V(G) \rightarrow [0, 1]$ such that every clique $K$ in $G$ satisfies $\sum_{v\in K}f(v) \leq 1$ and every vertex $v\in V(G)$ demands $f(v) \leq 1/(d(v) + 1/2)$, then $G$ has a fractional coloring $\phi$ in which the measure of $\phi(v)$ for each vertex $v\in V(G)$ is at least $f(v)$. This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle