For a tree $T$, consider its smallest subtree $T^{\circ}$ containing all vertices of degree at least $3$. Then the remaining edges of $T$ lie on disjoint paths each with one endpoint on $T^{\circ}$. We show that the chromatic symmetric function of $T$ determines the size of $T^{\circ}$, and the multiset of the lengths of these incident paths. In particular, this generalizes a proof of Martin, Morin, and Wagner that the chromatic symmetric function distinguishes spiders.
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.
The distinguishing chromatic number of a graph $G$ is the smallest number of colors needed to properly color the vertices of $G$ so that the trivial automorphism is the only symmetry of $G$ that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4.
The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. The distinguishing chromatic index $χ'_D (G)$ of a graph $G$ is the least number $d$ such that $G$ has a proper edge labeling with $d$ labels that is preserved only by the identity automorphism of $G$. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.
The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $Γ(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-Ω(\log n)}$ we have, $χ_D(Γ)\leχ(Γ) + 1$.
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $χ_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $Δ(G)$, then $χ_{D}(G)\leq Δ(G)+1$. We also obtain an upper bound for $χ_{D}(G)$ where $G$ is a graph with at most one cycle. Finally, we state a relationshi…
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $Δ$ and with no isolated edges has an \avd-coloring with at most $Δ+300$ colors, provided that $Δ>10^{20}$.
The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.
A graph $G$ is Ramsey for a graph $H$ if every colouring of the edges of $G$ in two colours contains a monochromatic copy of $H$. Two graphs $H_1$ and $H_2$ are Ramsey equivalent if any graph $G$ is Ramsey for $H_1$ if and only if it is Ramsey for $H_2$. A graph parameter $s$ is Ramsey distinguishing if $s(H_1)\neq s(H_2)$ implies that $H_1$ and $H_2$ are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.
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…
Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In fact, we prove that chi D(G) \leq chi L(G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with the property that chi D(G)= chi L(G) = 3.
A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges receive different colours and every vertex and incident edge receive different colours. This paper considers a strengthening of this condition and examines the minimum number of colours required for a total colouring with the additional property that for any adjacent vertices $u$ and $v$, the sets of colours incident to $u$ is different from the set of colours i…
Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of $T$ by their numbers of edges and leaves. We prove that $S_T = <Φ,X_T>$, where $<\cdot,\cdot>$ is the Hall inner product on symmetric functions and $Φ$ is a certain symmetric function that does not depend on $T$. Thus the chromatic symmetric function is a stronge…
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factor…
Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. Finck characterized the class of graphs that satisfy equality in this bound. In this paper, we provide a new characterization of this class of graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of graphs in this class. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromati…
We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $χ_D(G)$, and the distinguishing chromatic index $χ'_D(G)$, for vertex and edge colourings, respectively. We show that $χ_D(G)\leq 2Δ(G)-1$ for graphs with a finite maximum degree $…
Erdős proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishes nonisomorphic trees with the same number of vertices, is still open in general. For special trees it has already been shown. In 2008, Martin, Morin and Wagner proved it for spiders and some caterpillars. We decompose a tree by separating the leafs and their neighbors and do the same to the remai…
The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K_2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matousek and Ziegler who recognized a hierarchy of the different topological e…
The chromatic sum $Σ(G)$ of a graph $G$ is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present $Σ(G)<χ_f(G)|G|$ for every graph $G$.
In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most $5$, and for the $p$th chromatic number $χ_p$, from which follows in particular that graphs $G$ of large girth excluding a minor have $χ_p(G)\leq p+2$.
In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Δ(G)$, chromatic index $χ'(G)$ and total chromatic number $χ''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ'…
DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvořák and Postle. Several known bounds for the list chromatic number of a graph $G$, $χ_\ell(G)$, also hold for the DP-chromatic number of $G$, $χ_{DP}(G)$. On the other hand, there are several properties of the DP-chromatic number that shows that it differs with the list chromatic number. In this note we show one such property. It is well known that $χ_\ell (K_{k,t}) = k+1$ if and only if $t \geq k^k$. We show that $χ_{DP} (K_{k,t}) = k+1$ if $t \geq 1 + (k^k/k!)(\log(k!)+1)$, and we…
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 $G$.
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
A {\em chromatic root} is a root of the chromatic polynomial of a graph. While the real chromatic roots have been extensively studied and well understood, little is known about the {\em real parts} of chromatic roots. It is not difficult to see that the largest real chromatic root of a graph with $n$ vertices is $n-1$, and indeed, it is known that the largest real chromatic root of a graph is at most the tree-width of the graph. Analogous to these facts, it was conjectured in [8] that the real parts of chromatic roots are also bounded above by both $n-1$ and the tree-width of the graph. In thi…
A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, …
Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting b…
Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of gra…
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 …
The Grundy number of a graph $G$ is the maximum number of colors used by the First-Fit coloring of $G$ and is denoted by $Γ(G)$. Similarly, the ${\rm b}$-chromatic number ${\rm{b}}(G)$ of $G$ expresses the worst case behavior of another well-known coloring procedure i.e. color-dominating coloring of $G$. We obtain some families of graphs $\mathcal{F}$ for which there exists a function $f(x)$ such that $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. Call any such family $(Γ,b)$-bounded family. We conjecture that the family of ${\rm b}$-monotone graphs is $(Γ,b)$-bounded and valid…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $χ_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $χ_{st}^{'} (G )\leq \lfloor\frac{3Δ}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $Δ\geq 3.$ In this paper we obtained that $χ_{st}^{'}(G) \leq Δ+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph wit…
The distinguishing chromatic number, $χ_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $φ$, of $G$, such that the only automorphism of $G$ that preserves all colors of $φ$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $χ_D(G)\le Δ+1$. We prove this conjecture.
This note exhibits graphs whose chromatic polynomials have the Beraha number $B_{10} = (5 + \sqrt{5})/2$ as a root. It was previously known that no other non-integer Beraha number is a chromatic root, and so these examples complete the determination of precisely which Beraha numbers can be chromatic roots.
In this note, we introduce a new method for constructing graphs with high chromatic number and small clique. Indeed, via this method, we present a new proof for the well-known Kneser's conjecture.
Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that ev…
In this note, we prove that for any integer $n\geq 3$ the b-chromatic number of the Kneser graph $KG(m,n)$ is greater than or equal to $2{\lfloor {m\over 2} \rfloor \choose n}$. This gives an affirmative answer to a conjecture of [6].
{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).
The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice-sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.
A note on vertex-critical induced subgraphs of shift graphs
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
In this note we obtain a new bound for the acyclic edge chromatic number $a'(G)$ of a graph $G$ with maximum degree $D$ proving that $a'(G)\leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].
In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.
Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained…
A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $χ\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $χ\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one …
