The 3-choosability of plane graphs of girth 4

A set S of vertices of the graph G is called k-reducible if the following is true: G is k-choosable if and only if G-S is k-choosable. A k-reduced subgraphH of G is a subgraph of G such that H contains no k-reducible set of some specific forms. In this paper, we show that a 3-reduced subgraph of a non-3-choosable plane graph G contains either adjacent 5-faces, or an adjacent 4-face and k-face, where k?6. Using this result, we obtain some sufficient conditions for a plane graph to be 3-choosable. In particular, if G is of girth 4 and contains no 5- and 6-cycles, then G is 3-choosable.     

Two spectral characterizations of regular, bipartite graphs with five eigenvalues

Graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure. We show that regular, bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles. This enables to determine, from the spectrum alone, the feasible families of numbers of common neighbors for each vertex with other vertices in its part. For particular spectra, such as [6,2⁹,0⁶,-2⁹,-6] (where exponents denote eigenvalue multiplicities), there is a unique such family, which makes it possible to characterize all graphs with this spectrum.Using this lemma we also to show that, for r?2, a graph has spectrum if and only if it is a graph of a 1-resolvable transversal design TD(r,r), i.e., if it corresponds to the complete set of mutually orthogonal Latin squares of size r in a well-defined manner.     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate (Ageev, 1999). In this paper, we prove that circle graphs with girth at least g ⩾ 5 contain a vertex of degree at most one, or a chain of g− 4 vertices of degree two, which implies Ageev's result in the case g = 5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g ⩾ 5 as well as a precise estimate of their maximum average degree.     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate [A.A. Ageev, Every circle graph with girth at least 5 is 3-colourable, Discrete Math. 195 (1999) 229-233]. In this paper, we prove that circle graphs with girth at least g≥5 and minimum degree at least two contain a chain of g−4 vertices of degree two, which implies Ageev’s result in the case g=5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g≥5 as well as a precise estimate of their maximum average degree.     

Cycles in graphs of uncountable chromatic number

We answer a question of Erdős [1], [2] by showing that any graph of uncountable chromatic number contains an edge through which there are cycles of all (but finitely many) lengths.     

The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

Explicit construction of regular graphs without small cycles

For every integerd>2 we give an explicit construction of infinitely many Cayley graphsX of degreed withn(X) vertices and girth >0.4801...(logn(X))/log (d−1)−2. This improves a result of Margulis. Dedicated to Paul Erdős on his seventieth birthday     

On the hole index of L(2,1)-labelings of r-regular graphs

An L(2,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G so that adjacent vertices get labels at least distance two apart and vertices at distance two get distinct labels. A hole is an unused integer within the range of integers used by the labeling. The lambda number of a graph G, denoted λ(G), is the minimum span taken over all L(2,1)-labelings of G. The hole index of a graph G, denoted ρ(G), is the minimum number of holes taken over all L(2,1)-labelings with span exactly λ(G). Georges and Mauro [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] conjectured that if G is an r-regular graph and ρ(G)?1, then ρ(G) must divide r. We show that this conjecture does not hold by providing an infinite number of r-regular graphs G such that ρ(G) and r are relatively prime integers.     

ON 3-CHOOSABILITY OF PLANE GRAPHS WITHOUT 6-,7- AND 9-CYCLES

The choice number of a graph G,denoted by X1(G),is the minimum number k such that if a list of k colors is given to each vertex of G,there is a vertex coloring of G where each vertex receives a color from its own list no matter what the lists are. In this paper,it is showed that X1 (G)≤3 for each plane graph of girth not less than 4 which contains no 6-, 7- and 9-cycles.     

Decomposing graphs with girth at least five under degree constraints

We prove that the vertex set of a simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2. © 2000 John Wiley & Sons, Inc: J Graph Theory 33: 237–239, 2000     

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph $G$ with girth at least $5$ is packable.The conjecture was proved by Faudree et al. with the additional condition that $G$ has at most $\frac{6}{5}n?2$ edges. In this paper, for each integer $k\ge 3$, we prove that every non-star graph with girth at least $5$ and at most $\frac{2k?1}{k}n?{\alpha }_k\left(n\right)$ edges is packable, where ${\alpha }_k\left(n\right)$ is $o\left(n\right)$ for every $k$. This implies that the conjecture is true for sufficiently large planar graphs.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

On multiplicative graphs and the product conjecture

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Ne?et?il and Pultr These results allow us (in conjunction with earlier results of Ne?et?il and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].     

The forcing number of graphs with given girth

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The forcing number, originally known as the zero forcing number, and denoted F (G), of G is the cardinality of a smallest forcing set of G. We study lower bounds on the forcing number in terms of its minimum degree and girth, where the girth g of a graph is the length of a shortest cycle in the graph. Let G be a graph with minimum degree δ ≥ 2 and girth g ≥ 3. Davila and Kenter [Theory and Applications of Graphs, Volume 2, Issue 2, Article 1, 2015] conjecture that F (G) ≥ δ + (δ ? 2)(g ? 3). This conjecture has recently been proven for g ≤ 6. The conjecture is also proven when the girth g ≥ 7 and the minimum degree is sufficiently large. In particular, it holds when g = 7 and δ ≥ 481, when g = 8 and δ ≥ 649, when g = 9 and δ ≥ 30, and when g = 10 and δ ≥ 34. In this paper, we prove the conjecture for g ∈ {7, 8, 9, 10} and for all values of δ ≥ 2.     

An oriented coloring of planar graphs with girth at least five

An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with a maximum average degree less than and girth at least 5 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least 5 has an oriented chromatic number at most 16, that improves the previous known bound of 19 due to Borodin et al. [O.V. Borodin, A.V. Kostochka, J. Nešet?il, A. Raspaud, É. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999) 77-89].     

Reductions of 3-connected graphs with minimum degree at least four

We show that if G is a 3-connected graph of minimum degree at least 4 and with |V (G)| ≥ 7 then one of the following is true: (1) G has an edge e such that G/e is a 3-connected graph of minimum degree at least 4; (2) G has two edges uv and xy with ux, vy, vx ∈ E(G) such that the graph G/uv/xy obtained by contraction of edges uv and xy in G is a 3-connected graph of minimum degree at least 4; (3) G has a vertex x with N(x) = {x₁, x₂, x₃, x₄} and x₁x₂, x₃x₄ ∈ E(G) such that the graph (G ? x)/x₁x₂/x₃x₄ obtained by contraction of edges x₁x₂ and x₃x₄ in G – x is a 3-connected graph of minimum degree at least 4.

Each of the three reductions is necessary: there exists an infinite family of 3- connected graphs of minimum degree not less than 4 such that only one of the three reductions may be performed for the members of the family and not the two other reductions.     

Graphs with prescribed degree sets and girth

For a finite nonempty set of integers, each of which is at least two, and an integern 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.Research of the third author was partially supported by a Faculty Research Fellowship from Western Michigan University.     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.     

Q-integral graphs with edge-degrees at most five

We consider the problem of determining the Q-integral graphs, i.e. the graphs with integral signless Laplacian spectrum. We find all such graphs with maximum edge-degree 4, and obtain only partial results for the next natural case, with maximum edge-degree 5.