List version of [formula omitted]-labelings

Choosability with union separation of planar graphs without cycles of length 4

Let G = G(n) be a graph on n vertices with maximum degree Δ =Δ (n). Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as n →∞) of the existence of a proper coloring φ of G, such that for every vertex v of G, a so‐called L‐coloring. We give various lower bounds on σ, in terms of n, k, and Δ, which ensures that with probability tending to 1 as n →∞ there is an L‐coloring of G. In particular, we show, for all fixed k and growing n, that if and , then the probability that G has an L‐coloring tends to 1 as . If and , then the same conclusion holds provided that . We also give related results for other bounds on Δ, when k is constant or a strictly increasing function of n.

For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability puts $ k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k $ copies of $1$, then $\lambda$-choosability is equivalent to $k $-colourability. %We say $G$ is non-chromatic-$\lambda$-choosable if $\chi(G)=k_{\lambda}$ and $G$ is not $\lambda$-choosable. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

A proper vertex colouring of a graph G is 2-frugal (resp. linear) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph G is 2-frugally (resp. linearly) L-colourable if for a given list assignment $${L:V(G)\mapsto 2^{\mathbb N}}$$, there exists a 2-frugal (resp. linear) colouring c of G such that $${c(v) \in L(v)}$$ for all $${v\in V(G)}$$. If G is 2-frugally (resp. linearly) L-list colourable for any list assignment such that |L(v)| ≥ k for all $${v\in V(G)}$$, then G is 2-frugally (resp. linearly) k-choosable. In this paper, we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree.

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable if for a given list assignment L = {L(v): v e V}, there exists a proper coloring c of G such that c (v) e L(v) for all v e V. If G is L-list colorable for every list assignment with |L (v)| ≥ k for all v e V, then G is said k-choosable. A graph is said to be acyclically k-choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k-choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281300, 2006

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n − 1)k + 4k(n − 4) + 2·3k − 2((δ + 1)k − δk)(3n − 6 − m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0⩽3n − 6 − m≤⌊n/2⌋ − 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112-123, 2011 © 2011 Wiley Periodicals, Inc.

Locally self-avoiding Eulerian tours

Partitions of graphs and multigraphs under degree constraints

Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).

The distance Roman domatic number of a graph

Total labellings, irregularity strengths and colourings

Given a simple graph G, m a positive integer. The square of path graph P_m, denoted by P_m^2, is a graph obtained from P_m by adding new edges between any pair of vertices at distance at most 2 in P_m. A graph G is P_m^2-saturated if G does not contain P_m^2 as a subgraph, but the addition of any edge between two nonadjacent vertices in G contain P_m^2. The minimum size of P_m^2-saturated graph on n vertices is called a saturation number for P_m^2, denoted by sat(n,P_m^2). A set Sat(n,P_m^2 )={G:|V(G)|=sat(n,P_m^2) and G a P_m^2-saturated graph}. All graphs in Sat(n,P_m^2) are obtained computationally for n≤8 and m≤8 and expressed by their degree sequence.

Let \(G\) be a simple graph without isolated vertices with vertex set \(V(G)\) and edge set \(E(G)\) and let \(k\) be a positive integer. A function \(f:E(G)\longrightarrow \{-1, 1\}\) is said to be a signed star \(k\)-dominating function on \(G\) if \(\sum_{e\in E(v)}f(e)\ge k\) for every vertex \(v\) of \(G\), where \(E(v)=\{uv\in E(G)\mid u\in N(v)\}\). A set \(\{f_1,f_2,\ldots,f_d\}\) of signed star \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^df_i(e)\le k\) for each \(e\in E(G)\), is called a signed star \((k,k)\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed star \((k,k)\)-dominating family on \(G\) is the signed star \((k,k)\)-domatic number of \(G\), denoted by \(d^{(k,k)}_{SS}(G)\). In this paper we study properties of the signed star \((k,k)\)-domatic number \(d_{SS}^{(k,k)}(G)\). In particular, we present bounds on \(d_{SS}^{(k,k)}(G)\), and we determine the signed \((k,k)\)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.

Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall-Halmos-Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut-vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut-vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000