Non-isolated Vertices Research Articles

Topological and spectral properties of random digraphs.

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs of size n and connection probability p,D(n,p). In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices V_{x}(D) as well as two vertex-degree-based topological indices: the Randić index R(D) and sum-connectivity index χ(D). First, by performing a scaling analysis, we show that the average degree 〈k〉 serves as a scaling parameter for the average values of V_{x}(D),R(D), and χ(D). Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to (n,p), the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius sqrt[np(1-p)]. Subsequently, we compute six different invariants related to the eigenvalues of D(n,p) and observe that these quantities also scale with sqrt[np(1-p)]. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function (n,p). Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.

On odd colorings of sparse graphs

An oddc-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A proper conflict-freec-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free c-coloring is also an odd c-coloring. Cranston conjectured that every graph G with maximum average degree mad(G)<4cc+2 (where c≥4) has an odd c-coloring, and he proved this conjecture for c∈{5,6}. Note that the bound 4cc+2 is best possible. Cho et al. solved Cranston’s conjecture for c≥5, strengthening the result by transitioning from odd c-coloring to proper conflict-free c-coloring. However, they did not provide all the extremal non-colorable graphs G with mad(G)=4cc+2, which remains an open question of interest.In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free c-colorable graphs G with mad(G)=4cc+2. For the case of c=4, Cranston’s conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a 5-cycle. Cho et al. proved that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. We improve this result by proving that a graph G with mad(G)≤229 (with equality allowed) is not odd 4-colorable if and only if G belongs to a specific class of graphs. On the other hand, Cho et al. established that a planar graph with girth at least 5 has an odd 6-coloring; we improve it by proving that a planar graph without 4−-cycles adjacent to 7−-cycles also has an odd 6-coloring.

On odd colorings of planar graphs

An odd k-coloring of a graph G is a proper k-coloring such that every non-isolated vertex in G has a color appearing odd times on its neighborhood. In 2022, Petruševski and Škrekovski [Colorings with neighborhood parity condition, Discrete Appl. Math., 321: 385-391] introduced this concept and conjectured that every planar graph admits an odd 5-coloring. We prove that (1) every planar graph without adjacent 3-cycles is odd 7-colorable; (2) If G is a planar graph without 3-cycles and intersecting 4-cycles, then G is odd 5-colorable; (3) The odd chromatic number of a Halin graph is either 3 or 4.

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in its (open) neighborhood. Typical examples of graphs with large proper conflict-free chromatic number include graphs with large chromatic number and bipartite graphs isomorphic to the 1-subdivision of graphs with large chromatic number. In this paper, we prove two rough converse statements that hold even in the list-coloring setting. The first is for sparse graphs: for every graph H, there exists an integer cH such that every graph with no subdivision of H is (properly) conflict-free cH-choosable. The second applies to dense graphs: every graph with large conflict-free choice number either contains a large complete graph as an odd minor or contains a bipartite induced subgraph that has large conflict-free choice number. These give two incomparable (partial) answers of a question of Caro, Petruševski and Škrekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about proper conflict-free coloring and odd coloring in the literature. Moreover, we prove that every graph with layered treewidth at most w is (properly) conflict-free (8w−1)-choosable. This result applies to (g,k)-planar graphs, which are graphs whose coloring problems have attracted attention recently.

Partial domination and irredundance numbers in graphs

A dominating set of a graph G=(V(G),E(G)) is a vertex set D such that every vertex in V(G)∖D is adjacent to a vertex in D. The cardinality of a smallest dominating set of G is called the domination number of G and is denoted by γ(G). A vertex set D is a k-isolating set of G if G−NG[D] contains no k-cliques. The minimum cardinality of a k-isolating set of G is called the k-isolation number of G and is denoted by ιk(G). Clearly, γ(G)=ι1(G). A vertex set I is irredundant if, for every non-isolated vertex v of G[I], there exists a vertex u in V(G)∖I such that NG(u)∩I={v}. An irredundant set I is maximal if the set I∪{u} is no longer irredundant for any u∈V(G)∖I. The minimum cardinality of a maximal irredundant set is called the irredundance number of G and is denoted by ir(G). Allan and Laskar [1] and Bollobás and Cockayne [2] independently proved that γ(G)<2ir(G), which can be written ι1(G)<2ir(G), for any graph G. In this paper, for a graph G with maximum degree Δ, we establish sharp upper bounds on ιk(G) in terms of ir(G) for Δ−2≤k≤Δ+1.

1-planar graphs are odd 13-colorable

An odd coloring of a graph G is a proper coloring such that any non-isolated vertex in G has a color appearing an odd number of times on its neighbors. The odd chromatic number, denoted by χo(G), is the minimum number of colors that admits an odd coloring of G. Petruševski and Škrekovski in 2021 introduced this notion and proved that if G is planar, then χo(G)≤9 and conjectured that χo(G)≤5. More recently, Petr and Portier improved 9 to 8. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showed that every 1-planar graph is odd 23-colorable. In this paper, we improved this result and showed that every 1-planar graph is odd 13-colorable.

The Odd Chromatic Number of a Planar Graph is at Most 8

Petruševski and Škrekovski recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph G is said to be odd if for each non-isolated vertex x in V(G) there exists a colour c appearing an odd number of times in its neighbourhood N(x). Petruševski and Škrekovski proved that for any planar graph G there is an odd colouring using at most 9 colours and, together with Caro, showed that 8 colours are enough for a significant family of planar graphs. We show that 8 colours suffice for all planar graphs.

The odd chromatic number of a toroidal graph is at most 9

The odd coloring is a NP-hard problem in graph theory research. It is a proper coloring with an additional restriction that every non-isolated vertex has some colors that appear an odd number of times in its neighborhood. In a graph G, we use χo(G) to represent the minimum number of colors that ensures an odd coloring. A graph G is odd k-colorable if χo(G)≤k. The above definition about odd coloring was introduced very recently by Petruševski and Škrekovski, who also proved that if G is planar then χo(G)≤9. A toroidal graph is a graph that can be embedded on a torus. In this paper, we study the odd chromatic number of toroidal graphs and prove that every toroidal graph G has a property that χo(G)≤9. Since K7 is a toroidal graph, χo(G) is no less than 7.

Odd coloring of sparse graphs and planar graphs

An odd c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph G with mad(G)≤4c−4c+1 has an odd c-coloring for c≥4, and proved the conjecture for c∈{5,6}. In particular, planar graphs with girth at least 7 and 6 have an odd 5-coloring and an odd 6-coloring, respectively.We completely resolve Cranston's conjecture. For c≥7, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for c=4, we construct counterexamples, which all contain 5-cycles. On the other hand, we show that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. This implies that a planar graph with girth at least 11 has an odd 4-coloring. We also prove that a planar graph with girth at least 5 has an odd 6-coloring.

Directed random geometric graphs: structural and spectral properties

In this work we analyze structural and spectral properties of a model of directed random geometric graphs: given n vertices uniformly and independently distributed on the unit square, a directed edge is set between two vertices if their distance is smaller than the connection radius , which is randomly drawn from a Pareto distribution. This Pareto distribution is characterized by the power-law decay α and the lower bound of its support ; thus the graphs depend on three parameters . By increasing , for fixed , the model transits from isolated vertices () to complete graphs (). We first propose a phenomenological expression for the average degree which works well for α > 3, when k is a self-averaging quantity. Then we numerically demonstrate that , for all α, where is the number of nonisolated vertices of G. Finally, we explore the spectral properties of by the use of adjacency matrices represented by diluted random matrix ensembles; a non-Hermitian and a Hermitian one. We find that is a good scaling parameter of spectral and eigenvector properties of G mainly for large α.

Colorings with neighborhood parity condition

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring φ of a graph G is said to be odd if for each non-isolated vertex x ∈ V ( G ) there exists a color c such that φ − 1 ( c ) ∩ N ( x ) is odd-sized. We prove that every simple planar graph admits an odd 9-coloring, and conjecture that 5 colors always suffice.

Remarks on odd colorings of graphs

A proper vertex coloring φ of graph G is said to be odd if for each non-isolated vertex x∈V(G) there exists a color c such that φ−1(c)∩N(x) is odd-sized. The minimum number of colors in any odd coloring of G, denoted χo(G), is the odd chromatic number. Odd colorings were recently introduced in Petruševski and Škrekovski (0000). Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to degenericity or maximum degree, and pose several questions and problems.

Remarks on proper conflict-free colorings of graphs

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χpcf(G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χpcf(G) in terms of other graph parameters. In particular, we show that χpcf(G)≤5Δ(G)/2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4≤k≤6. The paper concludes with few open problems.

Irredundance trees of diameter 3

A set D of vertices of a graph G=(V,E) is irredundant if each non-isolated vertex of G[D] has a neighbour in V−D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G).The IR-graph of G has the IR(G)-sets as vertex set, and sets D and D′ are adjacent if and only if D′ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in D′. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K1,2n.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

Nonuniform random graphs on the plane: A scaling study.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree 〈k〉 and the number of nonisolated vertices V_{×}, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for 〈k(n,σ,ℓ)〉. Then, we look for the scaling properties of the normalized average measure 〈X[over ¯]〉 (where X stands for V_{×}, r, and S) over graph ensembles. We demonstrate that the scaling parameter of 〈V_{×}[over ¯]〉=〈V_{×}〉/n is indeed 〈k〉, with 〈V_{×}[over ¯]〉≈1-exp(-〈k〉). Meanwhile, the scaling parameter of both 〈r[over ¯]〉 and 〈S[over ¯]〉 is proportional to n^{-γ}〈k〉 with γ≈0.16.

Sombor index of directed graphs

Let D be a digraph with set of arcs A. The Sombor index of D is defined asSO(D)=12∑uv∈A(du+)2+(dv−)2,where du+ and dv− are the out-degree and in-degree of the vertices u and v of D. When D is a graph, we recover the Sombor index of graphs, a molecular descriptor recently introduced with a good predictive potential and a great research activity this year. In this paper we initiate the study of the Sombor index of digraphs. Specifically, we find sharp upper and lower bounds for SO over the class Dn of digraphs with n non-isolated vertices, the classes Cn and Sn of connected and strongly connected digraphs on n vertices, respectively, the class of oriented trees OT(n) with n vertices, and the class O(G) of orientations of a fixed graph G.

Extremal Graphs for Exponential VDB Indices

We Ąnd the extremal graphs for the exponential of well known vertexdegree-based topological indices over Gn, the set of graphs with n non-isolated vertices.

On the r-dynamic chromatic number of subdivision of wheel graph

Let H be simple, non-isolated vertex, connected and directionless graphs with vertex set V (H), edge set E(H). With the use of c-coloring of graph H, using the formula c : υ(H) → S, where |S| = c, the colors of the adjacent vertex are different. A proper c-coloring of H is an r-dynamic coloring if it’s applied to every vertex υ ∈ V (H) satisfies |c(N(υ))| ⩾ min {r, d(υ)}. The minimum c such that H has an r-dynamic coloring with c colors is called an r-dynamic chromatic number, denoted by χ r (H). In this study, we will obtain the exact value of r-dynamic chromatic number of subdivision by wheel graph.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.