Universal [formula omitted]-colorable graphs without a given configuration

This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

Extending precolorings to circular colorings

Let $$ \mathcal{H} $$ be a family of graphs. We say that G is $$ \mathcal{H} $$ -universal if, for each H ∈ $$ \mathcal{H} $$ , the graph G contains a subgraph isomorphic to H. Let $$ \mathcal{H} $$ (k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an $$ \mathcal{H} $$ (k, n)-universal graph Γ(k, n) with O(n 2 - 2/k (log n)1+8/k ) edges. This is optimal up to a small polylogarithmic factor, as Ω(n 2-2/k ) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ(k, n) we prove, using a probabilistic argument, that Γ(k, n) is $$ \mathcal{H} $$ (k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties.

Randomized online graph coloring

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of Ic is provided. We then construct a k-degenerate graph which is universal for the induced-hereditary property of finite k-degenerate graphs. In order to attempt the corresponding problem for the property of countable graphs with colouring number at most k + 1, the notion of a property with assignment is introduced and studied. Using this notion, we

Incidence coloring of Mycielskians with fast algorithm

Colouring graphs with bounded generalized colouring number

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

An L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

Given a graph $$G=(V,E)$$ , an edge bipartization set of G is a subset $$E'\subseteq E(G)$$ such that $$G'=(V,E{\setminus } E')$$ is bipartite. An edge bipartization set that is also a matching of G is called a bipartizing matching of G. Let $${\mathscr {BM}}$$ be the family of all graphs admitting a bipartizing matching. Although every graph has an edge bipartization set, the problem of recognizing graphs G having bipartizing matchings ( $$G\in \mathscr {BM}$$ ) is challenging. A (k, d)-coloring of a graph G is a k-coloring of V(G) such that each vertex of G has at most d neighbors with the same color as itself. Clearly a (k, 0)-coloring is a proper vertex k-coloring of G and, for any $$d>0$$ , the k-coloring is non-proper, also known as defective. A graph belongs to $$\mathscr {BM}$$ if and only if it admits a (2, 1)-coloring. Lovász (1966) proved that for any integer $$k>0$$ , any graph of maximum degree $$\varDelta $$ admits a $$\left( k,\lfloor \varDelta /k \rfloor \right) $$ -coloring. In this paper, we show that it is NP-complete to determine whether a 3-colorable planar graph of maximum degree 4 belongs to $$\mathscr {BM}$$ , i.e., (2, 1)-colorable. Besides, we show that it is NP-complete to determine whether planar graphs of maximum degree 6 or 8 admit a (2, 2) or (2, 3)-coloring, respectively. Due to Lovász (1966), our results are tight in the sense that on graphs with maximum degree three, five, and seven, there always exists a (2, 1), (2, 2), and (2, 3)-coloring, respectively. Finally, we present polynomial-time algorithms for particular graph classes, besides some remarks on the parameterized complexity of the problem of recognizing graphs in $${\mathscr {BM}}$$ .

Given any arbitrary nonempty subset [Formula: see text] of vertices in a graph [Formula: see text], each vertex [Formula: see text] in [Formula: see text] is associated with the vertex [Formula: see text] and is called an [Formula: see text]-open distance pattern coloring (odp coloring) of [Formula: see text] if no two adjacent vertices receive the same color [Formula: see text]. If such an [Formula: see text] exists, for a graph [Formula: see text], then [Formula: see text] is called an Open Distance Pattern Colorable (odpc) graph. The minimum cardinality of an odp coloring set in [Formula: see text], if it exists, is called the odpc-number of [Formula: see text] and is denoted by [Formula: see text]. Given any property [Formula: see text], we establish the characterization of odpc-graph with the property [Formula: see text]. Some classes of graphs are proved to be odpc. It is also proved that for a given positive integer [Formula: see text], there exists an odpc graph with an odp coloring set of cardinality [Formula: see text].

Oriented cliques and colorings of graphs with low maximum degree

Computing residual connectedness reliability for restricted networks

3-component domination numbers in graphs

For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ exists such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every pair of distinct vertices $u,v$ at distance at most $r$ in $G$. It was conjectured that $\chi'_{\Sigma,r}(G)\leq (1+o(1))\Delta^{r-1}$ for every $r\geq 3$. Thus far it has been in particular proved that $\chi'_{\Sigma,r}(G)\leq 6\Delta^{r-1}$ if $r\geq 4$. Combining probabilistic and constructive approach, we show that this can be improved to $\chi'_{\Sigma,r}(G)\leq (4+o(1))\Delta^{r-1}$ if the minimum degree of $G$ equals at least $\ln^8\Delta$.