James Davies
A wheel is a graph consisting of an induced cycle of length at least four and a single additional vertex with at least three neighbours on the cycle. We prove that no Burling graph contains an induced wheel. Burling graphs are triangle-free and have arbitrarily large chromatic number, so this answers a question of Trotignon and disproves a conjecture of Scott and Seymour.
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Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909).
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There are exactly four triangle-free graphs of order 9 which have 1-defective chromatic number 3. We show that these four graphs appear as subgraphs in almost all triangle-free graphs of order 10 with 1-defective chromatic numb…
More than twenty years ago Erdős conjectured~\cite{E1} that a triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of at least $k^{2 - \varepsilon}$ different lengths as $k \rightarrow \infty$. In this paper, we prove the stronger fact that every triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of $(\frac{1}{64} - \varepsilon)k^2 \log k$ consecutive lengths, and a cycle of length at least $(\tfrac{1}{4} - \varepsilon)k^2 \log k$. As there exist triangle-free graphs of chromatic number $k$ with at most roughly $4k^2 \log k$ v…
Let $G$ be any triangle-free graph with maximum degree $Δ\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of $G$, namely $χ_f(G)\leq 14/5. Recently, Hatami and Zhu proved $χ_f(G) \leq 3 -{3/64}$. In this paper, we prove $χ_f(G) \leq 3- 3/43$.
A tamed family of triangle-free graphs with unbounded chromatic number
构造了一个具有无界色数的无三角形图遗传类，其中每个非平凡图含不相邻孪生顶点或大小至多2的无边顶点割集，反面回答了Chudnovsky等人的问题，并证明twincut图是边临界的。
A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number $χ_k(G)$ is the least positive integer $m$ for which graph G is (m,k)-colourable. Let f(m,k;tfp) be the smallest order of a triangle-free planar graph such that $χ_k(G)$=m. In this paper we show that f(3,1;tfp)=11.
A connected $k$-chromatic graph $G$ with $k \geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B. Toft posed the problem of showing that the complete graphs on more than two vertices are the only triangle-critical graphs. By applying a method of M. Stiebitz [Discrete Math. 64 (1987), 91--93], we answer the problem affirmatively for triangle-critical $k$-chromatic graphs with $k \leq 6$.
Some results on chromatic number as a function of triangle count
本文研究色数作为顶点数、边数和三角形数量函数的上界，结果平滑插值于一般图与无三角形图的界之间，且在多数情况下是紧的，并指出一个开放问题可解决剩余差距。
Suns in triangle-free graphs of large chromatic number
研究三角自由图中色数较大时是否存在诱导子图为特定太阳结构，证明色数至少为48的图存在特定太阳结构或近似结构。
A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 24 vertices. This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that the smallest regular triangle-free 5-chromatic graphs have 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-chro…
Triangle-free subgraphs with large fractional chromatic number
本文证明了 Erdős-Hajnal 猜想中围长为4情形的分数色数版本，即存在函数 h(k,4)，使色数足够大的图必含分数色数至少为 k 的无三角形子图。
We prove that every subcubic triangle-free graph has fractional chromatic number at most 14/5, thus confirming a conjecture of Heckman and Thomas [A new proof of the independence ratio of triangle-free cubic graphs. Discrete Math. 233 (2001), 233--237].
We prove that for every $n$, there is a graph $G$ with $χ(G) \geq n$ and $ω(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $ω(H) \leq 2$ satisfies $χ(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
The chromatic profile of locally bipartite graphs
本文研究局部二部图的色轮廓，证明最小度超过4/7·n时必为3可着色且该界紧，超过6/11·n时必为4可着色。
An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $χ_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $χ_f(G) \leq 3-3/64 \approx 2.953$. Lu and …
Sparse chromatic graphs and the complete-graph triangle bound
本文证明稀疏色图（边数至多 $cr^3\log^2 r$）必含 $\binom r3$ 个三角形，通过稀疏核、阶均匀稳定性和稠密公共调色板列表方法给出完整证明。
For $d > 0$, define $G(\mathbb{Q}^3, d)$ to be the graph whose set of vertices is the rational space $\mathbb{Q}^3$, where two vertices are adjacent if and only if they are a Euclidean distance $d$ apart. Let $χ(\mathbb{Q}^3, d)$ be the chromatic number of such a graph or, in other words, the minimum number of colors needed to color the points of $\mathbb{Q}^3$ so that no two points at distance $d$ apart receive the same color. An open problem, originally posed by Benda and Perles in the 1970s, asks if there exists $d$ such that $χ(\mathbb{Q}^3, d) = 3$. Through numerous efforts over the years…
Extremal triangle-free graphs with chromatic number at least four
本文将非二分无三角形图的 Erdős–Gallai/Andrásfai 边数极值结果推广到色数至少为四的情形，证明 $n\geq90$ 时边数至多为 $\lfloor(n-3)^2/4\rfloor+5$，并由 Grötzsch 图膨胀图证明其紧性。
11/4-colorability of subcubic triangle-free graphs
证明了除两个例外外，所有连通次立方无三角形图的分数色数不超过11/4，且该界紧，改进了平面图情形。研究问题为分数色数上界，方法为构造性证明，贡献在于紧界及例外刻画。
Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chroma…
Induced Subgraph Bounds on the Zero Forcing Number and Chromatic Consequences
研究关于图的色数、团数和零强制数的关系，通过无三角形诱导子图建立零强制数下界，提出\((χ, ω, Z)\)猜想，证明其对无三角形正则图成立并提供数值证据。
We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-\frac{1}{n+1/3}$.
A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.
Spectral extremal results for triangle-free graphs with chromatic number at least four
本文研究色数至少为四的三角图自由图的谱极值问题，确定了当n足够大时，G(2,4)是格罗茨sch图的吹胀图，并与另一研究结果一致。
In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.
Counting triangles in graphs with no wheels of order at least five
研究不含阶至少为4的轮图的图中三角形的最大数量，确定了极值结构并刻画了所有极图。
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.
Alon proved that for any graph $G$, $χ_\ell(G) = Ω(\ln d)$, where $χ_\ell(G)$ is the list chromatic number of $G$ and $d$ is the average degree of $G$. Dvořák and Postle recently introduced a generalization of list coloring, which they called correspondence coloring. We establish an analogue of Alon's result for correspondence coloring; namely, we show that $χ_c(G) = Ω(d/\ln d)$, where $χ_c(G)$ denotes the correspondence chromatic number of $G$. We also prove that for triangle-free $G$, $χ_c(G) = O(Δ/\ln Δ)$, where $Δ$ is the maximum degree of $G$ (this is a generalization of Johansson's resul…
Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs
本文证明了连通无三角形图（除五圈外）的色数上界为 $\lceil m/\sqrt{5.5n}\rceil+1$，并指出常数最优，等号在 Grötzsch 图和特定奇圈上成立。
Characterization and chromatic number of triangle-free graphs with diameter 2
本文修正了Shekarriz等人对直径为2的无三角形图的刻画中的小缺陷，并利用该刻画证明了关于这类图色数的若干结果。
We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Turán's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\leq \ell$ for any $\ell$ and any $k\geq 4$, again with a best possible density of $>(\frac{1}{4} -\varepsilon)n^2$ edges for $k\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the c…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $Δ(G) + 16$ color…
Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary P, the chromatic number is at most n^3/19.259+O(n^2).
(Theta, triangle)-free and (even hole, $K_4$)-free graphs. Part 1 : Layered wheels
提出分层轮构造，生成具有任意大树宽和周长的图，为通过诱导子图刻画大树宽图提供潜在结果，并在无（theta，三角形）图等经典图类中提供大树宽与大秩宽图的实例。
We prove that every triangle-free graph with maximum degree $Δ$ has list chromatic number at most $(1+o(1))\fracΔ{\ln Δ}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{Δ\ln\lnΔ}{\lnΔ}$.
Given graphs $T$ and $H$, the generalized Turán number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new v…
The chromatic profile of locally colourable graphs
本文研究局部 $b$-分图及团公共邻域可着色图的色度剖面，确定其色度阈值，并给出保证局部 $b$-分图为 $(b+1)$-可着色的最小度条件。
Geometric triangle-free graphs of large chromatic number
本文通过几何构造，在盒子、直线、圆等对象上构建无三角形且大围长的图族，显著提升色数下界，并解决相关猜想与问题。
Kostochka and Yancey proved that every $4$-critical graph $G$ has $e(G) \geq \frac{5v(G) - 2}{3}$, and that equality holds if and only if $G$ is $4$-Ore. We show that a question of Postle and Smith-Roberge implies that every $4$-critical graph with no $(7,2)$-circular-colouring has $e(G) \geq \frac{27v(G) -20}{15}$. We prove that every $4$-critical graph with no $(7,2)$-colouring has $e(G) \geq \frac{17v(G)}{10}$ unless $G$ is isomorphic to $K_{4}$ or the wheel on six vertices. We also show that if the Gallai Tree of a $4$-critical graph with no $(7,2)$-colouring has every component isomorphic…
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an arc for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence o…
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Jo…
We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjecture…
Triangle-free $d$-degenerate graphs have small fractional chromatic number
本文证明了Harris关于无三角形 $d$-退化图分数色数至多为 $(4+o(1))\frac{d}{\ln d}$ 的猜想，并作为推论验证了Esperet等人的猜想，同时提出了关于随机独立集的更一般结果。
The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$. The least value $k$ for which $G$ admits such labeling is called \emph{additive chromatic number} of $G$. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph $G$, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete …
Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $Ω\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a $2$-colored subgraph with chromatic number at least $n^{2/3}$, and this is best possible. We further show that for fixed positive integers $s,r$ with $s\leq r$, e…
In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number.
We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we determine the Gallai-Ramsey numbers for connected graphs with five vertices of chromatic number four.
Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $χ$ contains a rainbow independent set of size $\lceil\frac12χ\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $χ$ contains an induced cycle of length $Ω(χ\logχ)$ as $χ\to\infty$. Even if one only demands an induced path of length $Ω(χ\logχ)$…
A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in linear time. We further show that `5' is tight for every wheel graph with at least 5 vertices, and briefly discuss possible generalizations to planar graphs of treewidth 3.
In this paper a subset of High-Dimensional Random Apollonian networks, that we called Wheel Random Apollonian Graphs (WRAG), is considered. We show how to generate a Wheel Random Apollonian Graph from a wheel graph. We analyse some basic graph properties like vertices and edges cardinality, some question concerning cycles and the chromaticity in such type of graphs, we suggest further work on this type of graphs.
In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $χ$-bounded but we allow here the existence of $C_4$. The proof relies on the concept of Parity Changing Path, an adaptation of Trinity Changing Path which was recently introduced by Bonamy, Charbit and Thomassé to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.
Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$…
A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.
On the chromatic number of random triangle-free graphs
本研究针对边数为$Θ(n^{3/2}(\log n)^{1/2})$的随机无三角形图，通过与随机2-SAT、双Erdős–Rényi图着色问题的高概率等价性，确定了色数3→4、4→5转变的标度窗口宽度及后者的极限概率。
For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and chromatic number $Ω(n^{1/4})$, which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number $Θ(\log n)$. In addition, if we also require that our Hasse diagram has girth at least $k\geq 5$, we can achieve a chromatic number of at least $n^{\frac{1}{2k-3}+o(1)}$. These results have the following surprising geometric consequence. They imply the existence of a family $\mathcal{C}$ of $n$ curves in the plane such that the disjointness graph $G$ of $\mathcal{C}$ is…
Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very difficult to compute in general. As yet, still only precious few sharp results are known. In this paper, we obtain bounds on the Gallai-Ramsey number for wheels and the exact value for the wheel on $5$ vertices.
Let G be a triangle-free graph with maximum degree δ(G). We show that the chromatic number \c{hi}(G) is less than 67(1 + o(1))δ/ log δ.
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