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 expressions that can serve as lower bounds for the chromatic number of a graph.
Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r+2. This lower bound is tight in several cases by results in an earlier paper of the first two authors.
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.
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.
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…
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…
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…
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.
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 $…
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…
We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core.
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, …
A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping $λ$ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D $χ$ (G) and the distinguishing chromatic index D $χ$ (G) are defined similarly, with the additio…
We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which …
The Mycielskian construction, denoted $μ(G)$, takes a finite simple graph $G$ to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted $μ_t(G)$, takes $G$ to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of $μ(G)$ and $μ_t(G)$ in terms of the same parameters of $G$. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing.
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.
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 …
For studying topological obstructions to graph colorings, Hom-complexes were introduced by Lovász. A graph $T$ is called a test graph if for every graph $H$, the $k$-connectedness of $|Hom(T, H)|$ implies $χ(H)\geq k + 1 + χ(T)$. The proof of the famous Kneser conjecture is based on the fact that $\mathcal{K}_2$, the complete graph on $2$ vertices, is a test graph. This result was extended to all complete graphs by Babson and Kozlov. Their proof is based on generalized nerve lemma and discrete Morse theory. In this paper, we propose a new topological lower bound for the chromatic number of a s…
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. We say that a graph $G$ is $d$-distinguishing critical, if $D(G)=d$ and $D(H)\neq D(G)$, for every proper induced subgraph $H$ of $G$. This generalizes the usual definition of a $d$-chromatic critical graph. While the investigation of $d$-critical graphs is a well established part of coloring theory, not much is known about $d$-distinguishing critical graphs. In this paper we determine all $d$-distinguishing critical grap…
Given a proper total $k$-coloring $c:V(G)\cup E(G)\to\{1,2,\ldots,k\}$ of a graph $G$, we define the value of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of $G$, $χ"_Σ(G)$. Pilśniak and Woźniak (2013) conjectured that $χ"_Σ(G)\leq Δ(G)+3$ for any simple graph with maximum degree $Δ(G)$. In this paper, we prove this bound to be asymptotically correct by showing that $χ"_Σ(G)\leq Δ(G)(1+o(1))$. The main idea of our argument…
The union vertex-distinguishing chromatic index $χ'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges incident to each vertex is different. We prove that $χ'_\cup(G) \in \left\{ \left\lceil \log_2\left(n +1\right) \right\rceil, \left\lceil \log_2\left(n +1\right) \right\rceil+1 \right\}$ for a graph $G$ on $n$ vertices without a component of order at most two. This answers a question posed by Bousquet, Dailly, Duchêne, Kheddouci and Parreau, and independently by …
With any (not necessarily proper) edge $k$-colouring $γ:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $σ\_γ$ given by $σ\_γ(v)=\sum\_{e\ni v}γ(e)$.A neighbour-sum-distinguishing edge $k$-colouring is an edge colouring whose associated vertex colouring is proper.The neighbour-sum-distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admitsa neighbour-sum-distinguishing edge $k$-colouring.These notions naturally extends to total colourings of graphs that assign colours to both vertices and edges.We study in this paper equitable neighbo…
Inspired by the phenomenon of co-channel interference in communication network, a novel graph parameter, called $r$-vertex-strongly-distinguishing total coloring (abbreviate as $D(r)$-VSDTC), is proposed in this paper. Given a graph $G$, an $r$-VSDTC is an assignment of $k$ colors to $V(G)\cup E(G)$ such that any two adjacent or incident elements receive different colors and any two vertices with distance at most $r$ have distinct color-set, where the color-set of a vertex $u$ is the set of colors assigned on $u$ and its neighborhoods and incident edges. The \emph{$r$-vertex-strongly-distingui…
Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$ϕ(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where $N(x)=\{y\in V(G)|xy\in E(G)\}$. If $ϕ(x)\neq ϕ(y)$ for any edge $xy\in E(G)$, then $f$ is called a neighbor full sum distinguishing total $k$-coloring of $G$. The smallest value $k$ for which $G$ has such a coloring is called the neighbor full sum distinguishing total chromatic number of $G$ and denoted by fgndi$_{\sum}(G)$. The coloring is an extension of neighbor sum distinguishing…
Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerne…
For a graph $G$ and a positive integer $k$, a vertex labelling $f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial automorphism of $G$ preserves the sets $f^{-1}(i)$ for each $i\in\{1,\ldots,k\}$. The distinguishing chromatic number of a graph $G$, denoted $χ_D(G)$, is defined as the minimum $k$ such that there is a $k$-distinguishing labelling of $V(G)$ which is also a proper coloring of the vertices of $G$. In this paper, we prove the following theorem: Given $k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs $G_{i}=(V_i,E_i)$ such that …
