On the depth of a planar graph

In this paper, we have defined the concept of the depth of a planar graph. We show that, if G is a simple finite planar graph with p vertices and q edges and q > 3(p ? 1) ? p/2^s?1, then the depth of G is at least equal to s.     

Modularity of Erdős‐Rényi random graphs

For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The modularity q^?(G) of the graph G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q^?(G)<1. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erd?s‐Rényi random graph G_n,p with n vertices and edge‐probability p. Two key findings are that the modularity is 1+o(1) with high probability (whp) for np up to 1+o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)^?1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

On the chromaticity of the 2-degree integral subgraph ofq-trees

A graphG is called to be a 2-degree integral subgraph of aq-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactlyq- 1 triangles. An added-vertexq-treeG with n vertices is obtained by taking two verticesu, v (u, v are not adjacent) in a q-treesT withn -1 vertices such that their intersection of neighborhoods ofu, v forms a complete graphK _q , and adding a new vertexx, new edgesxu, xv, xv ₁,xv ₂, …,xv _{q- 4}, where {v ₁,v ₂,...,v _q?4} ?-K _q . In this paper we prove that a graphG with minimum degree not equal toq -3 and chromatic polynomialP(G;λ) = λ(λ - 1) … (λ -q +2)(λ -q +1)³(λ -q) ^{n- q- 2} withn ≥ q + 2 has and only has 2-degree integral subgraph of q-tree withn vertices and added-vertex q-tree withn vertices.     

Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ? cd (G). We determine the graph Δ(G) for the finite simple groups of types A _?(q) and ² A _? (q ²), that is, for the simple linear and unitary groups.     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.     

On a conjecture by Plummer and Toft

The cyclic chromatic number χ_c(G) of a 2‐connected plane graph G is the minimum number of colors in an assigment of colors to the vertices of G such that, for every face‐bounding cycle f of G, the vertices of f have different colors. Plummer and Toft proved that, for a 3‐connected plane graph G, under the assumption Δ*(G) ≥ 42, where Δ*(G) is the size of a largest face of G, it holds that χ_c(G) ≤ Δ*(G) + 4. They conjectured that, if G is a 3‐connected plane graph, then χ_c>(G) ≤ Δ*(G) + 2. In the article the conjecture is proved for Δ*(G) ≥ 24. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 177–189, 1999     

On the Size of Graphs of Class 2 Whose Cores have Maximum Degree Two

The core G _Δ of a graph G is the subgraph of G induced by the vertices of maximum degree Δ(G). In this paper, we show that if G is a connected graph with Δ(G _Δ) ≤ 2 and ${\Delta(G)\ge\frac12(|V(G)|-1)}$ , then G is of class 2 if and only if G is overfull. Our result generalizes several results of Hilton and Zhao.     

On generalized Petersen graphs labeled with a condition at distance two

An L(2,1)-labeling of graph G is an integer labeling of the vertices in V(G) such that adjacent vertices receive labels which differ by at least two, and vertices which are distance two apart receive labels which differ by at least one. The λ-number of G is the minimum span taken over all L(2,1)-labelings of G. In this paper, we consider the λ-numbers of generalized Petersen graphs. By introducing the notion of a matched sum of graphs, we show that the λ-number of every generalized Petersen graph is bounded from above by 9. We then show that this bound can be improved to 8 for all generalized Petersen graphs with vertex order >12, and, with the exception of the Petersen graph itself, improved to 7 otherwise.     

On total 9‐coloring planar graphs of maximum degree seven

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Δ(G) is the maximum degree of G, then no graph has a total Δ‐coloring, but Vizing conjectured that every graph has a total (Δ + 2)‐coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Δ ≤ 7 has a total 9‐coloring by means of the discharging method. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 67–73, 1999     

Eigenvalues of a graph and its imbeddings

A graph H is imbedded in a graph G if a subset of the vertices of G determines a subgraph isomorphic to H. If λ(G) is the least eigenvalue of G and k_R(H) = lim sup_d→∞ {λ(G)| H imbedded in G; G regular and connected; diam(G) > d; deg(G) > d}, then λ(H) ? 2 ≤ k_R(H) ≤ λ(H) with these bounds being the best possible. Given a graph H, there exist arbitrarily large families of isospectral graphs such that H can be imbedded in each member of the family.     

仙人掌图的L（2，1）-标号

一个图G的L（2，1）-标号是给图G上的顶点分配非负整数标号，使得G上相邻的两个点的标号至少相差2，距离为2的两个点的标号则不同．G的L（2，1）-标号数λ（G）是所有能使图G正常标号的最小标号．如果一个图的任何两个圈不含有公共边，则称这个图为仙人掌图．显然树是它的一个子图类．对于任何树T，有△（T） ＋ 1 ≤λ（T） ≤△A（T）＋2．本文中我们证明了在一些条件下，这个界也适用于仙人掌图．     

On the chromaticity of certain subgraphs of a q-tree

We show that a graph G on n ? q + 1 vertices (where q ? 2) has the chromatic polynomial P(G;λ) = λ(λ ? 1) … (λ ? q + 2) (λ ? q + 1)² (λ ? q)^n?q?1 if and only if G can be obtained from a q-tree Ton n vertices by deleting an edge contained in exactly q ? 1 triangles of T. Furthermore, we prove that these graphs are triangulated.     

On q-trees

We show that a graph G with n vertices is a q-tree if and only if its chromatic polynomial is P(G, λ) = λ(λ – 1) ? (λ – q + 1) (λ – q)^n-q where n ≥q.     

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n,d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C_p,C_q of lengths p and q with p+q≡2(mod4). In this paper, we characterize the graphs with minimal energy in G(n,d).     

Line graph eigenvalues and line energy of caterpillars

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d?3 and n?2(d-1). Let p=[p₁,p₂,…,p_d-1] with p₁?1,p₂?1,…,p_d-1?1 such that

p₁+p₂+?+p_d-1=n-d+1.     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

On the Super Edge-Magic Deficiency of Graphs

A (p, q) graph G is edge-magic if there exists a bijective function f: V(G) ∪ E(G) → {1,2,…,p + q} such that f(u) + f(v) + f(uv) = k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V(G)) = {1,2,…,p}. The question studied in this paper is for which graphs is it possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic? If it is possible for a given graph G, then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, μ_s(G) of G; otherwise we define it to be + ∞.     

The 2-pebbling property for dense graphs

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f(G) is the smallest number m such that for every distribution of m pebbles and every vertex v,a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f(G) q pebbles, where q is the number of vertices with at least one pebble, it is possible,using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G(s,t) has the 2-pebbling property, where G(s, t) is a bipartite graph with partite sets of size s and t (s ≥ t). Similarly, the-pebbling number f (G) is the smallest number m such that for every distribution of m pebbles and every vertex v, pebbles can be moved to v. Herscovici et al. conjectured that f(G) ≤ 1.5n + 8-6 for the graph G with diameter 3, where n = |V (G)|. In this paper, we prove that if s ≥ 15 and G(s, t) has minimum degree at least (s+1)/ 2 , then f (G(s, t)) = s + t, G(s, t) has the 2-pebbling property and f (G(s, t)) ≤ s + t + 8(-1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G(s, t) with s ≥ 15 and minimum degree at least (s+1)/ 2 .