landon rabern
We outline some ongoing work related to a conjecture of Reed \cite{reed97} on $ω$, $Δ$, and $χ$. We conjecture that the complement of a counterexample $G$ to Reed's conjecture has connectivity on the order of $\log(|G|)$. We prove that this holds for a family (parameterized by $ε> 0$) of relaxed bounds; the $ε= 0$ limit of which is Reed's upper bound.
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Upper Chromatic Numbers: An Update
本文是2000年撰写的关于上色数的综述，概述了该领域的研究现状。
Problem of finding an optimal upper bound for a chromatic no. of 3K1-free graphs is still open and pretty hard. Here we find a tight chromatic upper bound for {3K1, C5}-free graphs. We prove that if G is {3K1, C5}-free, then the chromatic no. <= (3ω-1)/2 where ω is the size of a maximum clique in G and show with examples that the bound is tight.
Upper bounds for the list-distinguishing chromatic number
本文针对简单有限连通图，证明了Brooks定理在列表可区分色数上的类比，并利用着色数和列表色数给出两个上界，同时确定了书图等多种图族的列表可区分色数。
Let $G$ be a circle graph without clique on 4 vertices. We prove that the chromatic number of $G$ doesn't exceed 30.
Upper bounds on the odd graceful chromatic number of graphs
本文研究二部图的奇优美色数，给出若干仅依赖顶点数或其二部划分相关图色数的新上界。
Problem of finding an optimal upper bound for the chromatic no. of 3K1-free graphs is still open and pretty hard. It was proved by Choudum et al that an upper bound on the chromatic no. of {3K1, K1+C4}-free graphs, is 2ω. We improve this by proving that if G is {3K1, K1+C4}-free, then its chromatic no. is less than or equal to 3ω divided by 2, where ω is the size of a maximum clique in G. Also we give examples to show that this bound is tight.
A double-logarithmic upper bound on the chromatic number of the associahedron
本文证明了结合多面体的色数上界从 O(log n) 改进到 O(log log n)，主要贡献是给出了更紧的色数上界。
The multi-fold chromatic number of the plane $χ_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two corresponding $m$-subsets of $k$-set do not contain any common color. We consider upper bounds for $m$-fold chromatic numbers of the plane. Our main result is that for any $m$ the inequality $χ_m<(1+2/\sqrt3)^2\cdot m+3.501$ holds.
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane $\mathrm{PG}(2,q)$ for all $q$. In addition, we det…
Chromatic numbers for contact graphs of congruent cuboids
研究三维整数尺寸立方体接触图的色数，证明不允许旋转时色数上界为8，允许旋转时为48，并探讨了边长为1立方体的更精确上界。
This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If $a_r$ is the coefficient of the $q^r$ term in the chromatic polynomial $P(G,q)$, where $q$ is the number of colors, then we find $a_r \le {e \choose v-r} - {e-g+2 \choose v-r-g+2} + {e-k_g-g+2 \choose v-r-g+2} - \sum _{n=1}^{k_g-\ell_g}\sum_{m=1}^{\ell_g-1} {e-g+1-n-m \choose v-r-g} - δ_{g,3}\sum_{n=1}^{k_g+\ell_{g+1}^*-\ell_g} {e-\ell_g-g+1-n \choose v-r-g}$, where $k_g$ is the number of circuits of length $g$ and $\ell_g$ and $\ell_{g+1}^*$ ar…
We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the bound of Schauz on $k$-partite $k$-uniform hypergraphs. It makes sense for sparse hypergraphs: in particular we show that a $k$-uniform $k$-regular hypergraph has the list chromatic number 2 for $k \geq 4$. Also we obtain both lower and upper bound on the list chromatic number of a complete $s$-uniform 2-colorable hypergraph in the vein of Erd{\H o}s--Rubin--Taylor theorem.
Problem of finding an optimal upper bound for χ of (3 Times K1)-free graphs is still open and pretty hard. It was proved by Choudum et al that upper bound on the χ of {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free graphs is 2ω. We improve this by proving that if G is {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free, then χ less than or equal to 3ω divided by 2 for ω not equal to 5, and χ less than or equal to 8 for ω = 5 where ω is the size of a maximum clique in G. We also give examples of extremal graphs.
Upper chromatic number of generalized quadrangles
研究广义四边形超图的上色数，通过将点和线对应顶点和超边，定义彩虹着色与无彩虹着色，进而确定给定超图\(H\)的上色数\(\overline{\chi}(H)\)。
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that $a'(G)\le Δ+2$ for any simple graph $G$ with maximum degree $Δ$. In this paper, we prove that if $G$ is a planar graph, then $a'(G)\leqΔ+7$. This improves a result by Basavaraju et al. [{\em Acyclic edge-coloring of planar graphs}, SIAM J. Discrete Math., 25 (2011), pp. 463-478], which says that every planar graph $G$ satisfies $a…
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small …
Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.
The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 α\right \rceil$, where $α= α(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 χ(G) \right \rceil box(G) …
In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any linear extension of $Q_L$ generates a distinguishing 2-coloring of $L$. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.
In this paper we find chromatic numbers of distance graphs $G(n,3,2)$ for infinitely many n. Also we improve upper bound for $χ(G(n,r,s))$ in large part of cases.
Upper Bounds of the Odd Chromatic Number of a Graph in terms of its Thickness
本文研究图的奇色数上界，证明最小度至少为2倍厚度减1且围长至少为6的图可被6倍厚度种颜色奇染色。
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by $χ'_a(G)$. In this paper, we prove that $χ_a'(G)$ <= 5($Δ+2$)/2 for any graph G having maximum degree $Δ$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $χ_a'(G)$ <= 3$Δ$ for any graph G without is…
An $r$-dynamic $k$-coloring of a graph $G$ is a proper vertex $k$-coloring such that the neighbors of any vertex $v$ receive at least $\min\{r,{\rm deg}(v)\}$ different colors. The $r$-dynamic chromatic number of $G$, $χ_r(G)$, is defined as the smallest $k$ such that $G$ admits an $r$-dynamic $k$-coloring. In this paper we introduce an upper bound for $χ_r(G)$ in terms of $r$, chromatic number, maximum degree and minimum degree. In 2001, Montgomery \cite{MR2702379} conjectured that, for a $d$-regular graph $G$, $χ_2(G)-χ(G)\leq 2$. In this regard, for a $d$-regular graph $G$, we present two u…
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 …
The balanced upper chromatic number of linear hypergraphs and the $n$-cube over $t$ elements
本文研究线性超图与n-立方体$C_t^n$的平衡上色数，针对任意线性超图给出该参数的通用上界，确定$t\geq4n-2$时$C_t^n$的精确值，并在2、3维情形下给出更小t值时的相关结果。
In the present paper, we have found new upper bounds for chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete critical distance $d$ the chromatic number of $\Z^{n}$ with critical distance $\sqrt{2d}$ has a polynomial growth in $n$ with exponent less than or equal to $d$ (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any $l_{p}$ norm. Besides, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of $…
The detour order of a graph $G$, denoted $τ(G)$, is the order of a longest path in $G$. A partition $(A, B)$ of $V(G)$ such that $τ(\langle A \rangle) \leq a$ and $τ(\langle B \rangle) \leq b$ is called an $(a, b)$-partition of $G$. A graph $G$ is called $τ$-partitionable if $G$ has an $(a, b)$-partition for every pair $(a, b)$ of positive integers such that $a + b = τ(G)$. The well-known Path Partition Conjecture states that every graph is $τ$-partitionable. In \cite{df07} Dunber and Frick have shown that if every 2-connected graph is $τ$-partitionable then every graph is $τ$-partitionable. I…
On the exact quantum chromatic number of generalized Johnson graphs
本文研究广义约翰逊图的量子色数，通过构造正交表示、结合特征值分析确定两类无穷族的精确量子色数，发现其经典色数随n指数增长、量子色数线性增长，存在指数级分离。
A stronger upper bound on the D-chromatic index
该研究针对图的D-色指数的Brooks型上界问题，采用概率方法，首次改进了此前由贪心着色得到的上界，证明了最大度足够大时D-色指数不超过带正系数修正的二次上界。
The chromatic number $χ(\mathbb{R}^n)$ of the Euclidean space $\mathbb{R}^n$ is the smallest number of colors sufficient for coloring all points of the space in such a way that any two points at the distance 1 have different colors. In 1972 Larman--Rogers proved that $χ(\mathbb{R}^n) \leq (3 + o(1))^n$. We give a new proof of this bound.
We study the relation between the correspondence chromatic number and the Alon--Tarsi number, both upper bounds on the list chromatic number of a graph. There are many graphs with Alon--Tarsi number greater than the correspondence chromatic number. We present here a family of graphs with arbitrary Alon--Tarsi number, with correspondence chromatic number one larger. Keywords: correspondence coloring, Alon--Tarsi number AMS Mathematics Subject Classification: 05C15
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.
We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of t…
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.
A C-coloring of a hypergraph ${\cal H}=(X,{\cal E})$ is a vertex coloring $φ: X\to {\mathbb{N}}$ such that each edge $E\in{\cal E}$ has at least two vertices with a common color. The related parameter $\overlineχ({\cal H})$, called the upper chromatic number of ${\cal H}$, is the maximum number of colors can be used in a C-coloring of ${\cal H}$. A hypertree is a hypergraph which has a host tree $T$ such that each edge $E\in {\cal E}$ induces a connected subgraph in $T$. Notations $n$ and $m$ stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish …
Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $δ$, and maximum degree $Δ$. The "oriented chromatic number" of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which $δ\geq\log n$. We prove that every such graph has oriented chromatic number at least $Ω(\sqrt{n})$. In the case that $δ\geq(2+ε)\log n$, this lower bound is improved to $Ω(\sqrt{m})…
For a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $χ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $χ(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.
The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic number of random graphs. We show that with high probability $χ_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \mathbb{N}$. The upper bound applies for even $n$ and any other value of $p$ as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed…
The chromatic number of a latin square $L$, denoted $χ(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $χ(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture---commonly attributed to Brualdi---that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has ch…
Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We determine the rate of growth of of the sequence ${Q(n,\lceil rn \rceil)}_{n=1}^\infty$ for any fixed $0<r\leq 1$. We also give a better upper bound for $Q(n,\lceil rn \rceil)$.
On Sierpiński packing chromatic number and recognition of Sierpiński products
本文定义并研究了Sierpiński积图的堆积染色数及其上界，确定了完全图因子的情形，并证明路径或星因子的情形下染色数为3，且可多项式时间识别树因子表示。
Upper Bounds on the Chromatic Index of Linear Hypergraphs
本文研究线性超图的色指数上界，通过颜色保持群和轨道划分给出上界，并探讨Berge-Füredi猜想，提供三个基于Helly性质的充分条件。
For a multigraph G, the integer round-up phi(G) of the fractional chromatic index yields a good general lower bound for the chromatic index . For an upper bound, Kahn showed that for any real c > 0 there exists a positive integer N so that the chromatic index is less than (1+c)*phi(G) whenever the fractional index > N. We show the amount by which the chromatic index can surpass phi(G) is in fact logarithmic, by showing that for any multigraph G with order n > 3 and at least one edge, the chromatic index is less than phi(G) + log (min {(n+1)/3, phi(G)}) .
Chromatic numbers for facet colouring of some generalised associahedra
本文研究了几类广义关联多面体面着色的色数，给出了置换多面体、关联多面体和简单置换关联多面体的精确值，并建立了环面多面体和星形多面体的上下界及其渐近值。
A sequence is called non-repetitive if no of its subsequences forms a repetition (a sequence $r_1,r_2,\dots,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\leq i \leq n$). Let $G$ be a graph whose vertices are coloured. A colouring $φ$ of the graph $G$ is non-repetitive if the sequence of colours on every path in $G$ is non-repetitive. The Thue chromatic number, denoted by $π(G)$, is the minimum number of colours of a non-repetitive colouring of $G$. In this short note we present a general upper bound for the Thue chromatic number for the lexicographic product $G\circ H$ of graphs $G$ and $H$ with …
Given a weighted graph $G_\bx$, where $(x(v): v \in V)$ is a non-negative, real-valued weight assigned to the vertices of G, let $B(G_\bx)$ be an upper bound on the fractional chromatic number of the weighted graph $G_\bx$; so $χ_f(G_\bx) \le B(G_\bx)$. To investigate the worst-case performance of the upper bound $B$, we study the graph invariant $$β(G) = \sup_{\bx \ne 0} \frac{B(G_\bx)}{χ_f(G_\bx)}.$$ \noindent This invariant is examined for various upper bounds $B$ on the fractional chromatic number. In some important cases, this graph invariant is shown to be related to the size of the larg…
A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and the product of the chromatic number of a graph with that of its complement. In this paper, we provide a new characterization of the other class of graphs.
We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $χ'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $Δ(G) \le χ_g'(G) \le 2Δ(G)-1$ hold for every graph $G$, where $Δ(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, an…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we obtain tight upper bound $\left\lfloor\frac{3Δ}{2}\right\rfloor+2$ for the star chromatic index of every Halin graph, that proves the conjecture of Dvo{ř}{á}k et al. (J Graph Theory, 72 (2013), 313--326) for cubic Halin graphs.
Some coloring algorithms gives an upper bound for the locating chromatic number of trees with all the vertices not in an end-path colored by only two colors. That means, a better coloring algorithm could be achieved by optimizing the number of colors used in the end-paths. We provide an estimation of the locating chromatic number of trees using the locating chromatic number of its end-palms. We also study the locating chromatic number of palms, a subdivision of star. We also prove $χ_L(S_n(k))=Θ(n^{1/k})$; $χ_L(S_n(3))=(1+o(1))\sqrt[3]{4n}$; and $χ_L(O_n)=\left\lceil\log_3\left(\frac{n}{4}\rig…
Given a group $G$ of automorphisms of a graph $Γ$, the orbital chromatic polynomial $OP_{Γ,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $Γ.$ In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in $\mathbb{R}$, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in $[\frac{32}{27},\infty)$. Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic pol…
For an arbitrary invariant $ρ(G)$ of a graph $G$, the $ρ-$vertex stability number $vs_ρ(G)$ is the minimum number of vertices of $G$ whose removal results in a graph $H\subseteq G$ with $ρ(H)\neq ρ(G)$ or with $E(H)=\varnothing$. In this paper, first we give some general lower and upper bounds for the $ρ$-vertex stability number, and then study the edge chromatic stability number of graphs, $vs_{χ^{\prime}}(G)$, where $χ^{\prime}=χ^{\prime}(G)$ is edge chromatic number (chromatic index) of $G$. We prove some general results for this parameter and determine $vs_{χ^{\prime}}(G)$ for specific cla…
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. In this short note we present upper and lower bound for the packing chromatic number of the lexicographic product $G\circ H$ of graphs $G$ and $H$. Both bounds coincide in many cases. In particular this happens if $|V(H)|-α(H)\geq {\rm diam}(G)-1$, where $α(G)$ denotes the independence number of $G$.
Let $G$ be a graph. We use $χ(G)$ and $ω(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices, and an $HVN$ is a $K_4$ together with one more vertex which is adjacent to exactly two vertices of $K_4$. Combining with some known result, in this paper we show that if $G$ is $(P_5, \textit{HVN})$-free, then $χ(G)\leq \max\{\min\{16, ω(G)+3\}, ω(G)+1\}$. This upper bound is almost sharp.
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we first obtain some upper bounds for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of $2$-dimensional grids. We also obtain some upper bounds on the star chromatic index of the Cartesian product of a path with a cyc…
In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$.
Improved bounds for the chromatic index of $k$-uniform hypergraphs
研究k - 均匀t - 简单超图色指数上界，用概率和蚕食着色法证明一般定理，结构技术证特定情况b的上界，给出3 - 均匀2 - 简单和3 - 简单超图色指数上界推论。
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time. Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartesia…
Here we give refined numerical values for the minimum number of vertices of $k$-chromatic unit distance graphs in the Euclidean plane.
Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},τ+1)| \le (τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},τ+1)|/(τ-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast as $n \to \inft…
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