Graph coloring on paper: check every neighboring pair

Graph coloring on paper begins with a rule about connected points: give the ends of each connection different labels. In the original exercise here, five points form a closed cycle. Trying two labels gets you most of the way around, then exposes a conflict at the last connection. A third label resolves it, and a short check shows exactly why.

You can follow the exercise with letters or numbers alone. Red, blue and green are optional names for three labels. They make no claim about how someone sees color, which palette looks best or what an accessible display requires. Keep the written label beside each point so the mathematical assignment remains explicit even when you ignore the ink color.

Draw five points and define the connections

Put A near the top of your page, B toward the upper right, C toward the lower right, D toward the lower left and E toward the upper left. Leave enough space beside each point to write a label. These positions make a convenient drawing. The exercise does not attach a distance, angle or measured coordinate to any point.

Draw the connections AB, BC, CD, DE and EA. That is the complete connection list. There is no AC, AD, BD, BE or CE connection in this example. Keep the list next to the drawing so you can check a mark against the definition instead of deciding that two points must be connected because they look close together.

A point is a vertex and a connection is an edge in this graph. MathWorld describes vertex coloring as assigning colors to vertices so the endpoints of each edge receive different colors. That is the definition used here. The task colors points; a line simply identifies which pair must differ. Its thickness, length and direction on the page do not supply another requirement.

Try two labels around the open chain

Call the first label 1 and the second label 2. Assign A label 1. Since AB connects A to B, give B label 2. Next, BC requires C to differ from B. With only those two choices available, C must receive label 1. At this stage, both AB and BC satisfy the rule.

Continue to D. Its connection to C requires label 2, because C already has label 1. Then E must receive label 1 to differ from D along DE. Your assignment now reads A1, B2, C1, D2, E1. Write the full sequence before evaluating it. Keeping the proposal visible makes it possible to identify the particular connection that fails.

AB has labels 1 and 2. BC has 2 and 1. CD has 1 and 2, while DE has 2 and 1. Those four connections pass. The final connection EA has 1 at E and 1 at A. It fails the stated rule. Four successful checks do not make the fifth connection disappear.

Explain why changing the starting label cannot fix it

Perhaps starting A with label 2 would help. Work through that proposal on a fresh copy: A2 forces B1, then C2, D1 and E2. EA still joins equal labels, now 2 and 2. The names have switched while the closing conflict remains. This second attempt is easy to verify without erasing the first one.

These two cases cover both possible starting labels when only 1 and 2 are allowed. After choosing A, each successive point is forced by its connection to the previous point. There is no alternative choice at B, C, D or E that both differs from its predecessor and stays within the two-label set. Therefore neither starting choice yields a valid two-label assignment.

That argument concerns this exact five-point cycle with exactly the five declared edges. It does not rely on an impression that the drawing looks difficult. You have listed the starting cases and followed the consequences. If someone proposes a different two-label solution, inspect its entries against AB, BC, CD, DE and EA; at least one of those checks must expose the conflict described by the argument.

Add a third label and check all five edges

Return to A1, B2, C1 and D2. Change E to label 3. Now the complete assignment is A1, B2, C1, D2, E3. The first three edge checks remain unchanged: AB compares 1 with 2, BC compares 2 with 1, and CD compares 1 with 2. Each pair differs.

DE compares 2 at D with 3 at E. EA compares 3 at E with 1 at A. Those two pairs also differ. All five edges pass, so this assignment satisfies the rule. You can annotate each listed edge with its two labels and a check mark, keeping the verification separate from the drawing's appearance.

We have shown both that two labels cannot work and that three labels can work for this graph. Three is consequently its minimum number of labels. One label also fails immediately at AB, since both endpoints would receive it. The minimum conclusion comes from those explicit bounds, without needing a claim about every cycle or every graph.

Reuse a label when the graph allows it

A and C both carry label 1 in the successful assignment. That reuse is allowed because AC is absent from the declared edge list. Likewise, B and D both carry label 2, and BD is absent. Two points do not need different labels merely because they occupy different places on the sheet.

Read a proposed objection carefully. If it says A and C look nearby, return to the connection list. Nearness is not one of this exercise's constraints. If it says an AC line has been added, the exercise has changed and that new edge needs checking. Naming the change prevents a mark on a later sketch from silently becoming part of the original problem.

The same distinction applies when two drawn lines cross accidentally. A crossing mark does not introduce a new vertex unless you define it as one. For this exercise, retain the five named points and their five connections. Redraw a messy line if necessary so someone else can recover that same defined graph from the page.

When checking someone else’s proposal, read the edge list in its written order and record both endpoint labels before giving a verdict. A failed edge needs its own pair recorded, such as EA comparing 1 with 1. That note lets the other reader reproduce the disagreement. A bare comment that the picture looks wrong leaves the actual constraint unidentified.

Keep point names and labels separate

The letters A through E identify points. The numerals 1 through 3 identify labels assigned to them. Moving A's label from 1 to 2 would change the assignment, while renaming point A would change how you refer to its connections. Keeping those roles distinct makes a written correction easier to follow.

If you prefer color names, use red for 1, blue for 2 and green for 3. The successful assignment becomes A red, B blue, C red, D blue and E green. Write a small key with those equivalences. Renaming labels consistently preserves which pairs agree and which differ; the checks still use the same five edges.

Suppose an explicitly fictional learner, Noor, writes E green on the drawing but E1 in her edge-check notes. Her records describe different assignments. She can choose the intended one and update the other record before claiming a pass. This is a copying discrepancy, with no evidence that the underlying adjacency rule has changed.

Use the paper as a drawing surface

The Ampad quadrille pad bundle lists ten pads, with fifty white sheets per pad and 11-by-17-inch wording. Its native account describes quadrille ruling and printing on both sides. The sole gallery shows gridded paper. These are offer details, without an independent measurement or a tested ink-performance finding.

The separate Ampad legal quadrille pad bundle lists ten pads of forty sheets with 8.5-by-14-inch wording, glue binding and chipboard backing. Its sole gallery also shows gridded paper. Neither photograph independently counts the bundle or measures its ruling. Confirm any specification that matters to an actual order against that exact item.

The original exercise needs five distinguishable points and their declared connections. It does not depend on a printed square having a particular physical size. You can leave uneven gaps between vertices without changing the edge checks. Keeping the marks legible matters for reading your particular drawing; it supplies no product comparison or promise that either pad makes the exercise easier.

Separate adjacency from other grid questions

The equal area and different perimeter example counts interior unit squares and exposed boundary lengths for two defined rectangles. Its edges have a measurement role. Here, an edge records a pair of points whose labels must differ. No area or boundary-length total determines whether the coloring passes.

The graph paper grid-spacing guide separates sheet extent from the spacing of printed cells. That physical-paper distinction remains useful when choosing a sheet for another task. This exercise assigns no measured cell size and derives no paper specification from its drawing. The five-point definition and label list provide the information needed for its mathematical check.

Keep an optional purchase tied to its own information

If you consider buying a quadrille pad through republeq.com and prefer crypto such as USDT, identify the exact pad offer before deciding whether to authorize a purchase. The five-point exercise does not establish a sheet dimension, printed ruling or quantity in a delivered bundle. Read the current item account and ask about any specification that affects your choice. A payment preference supplies no missing paper information. For the actual basket, obtain the current transaction instructions and make sure you understand the requested asset, network, destination and order reference before acting. This article confirms no asset or network availability for a particular order and supplies no transfer address. If an instruction is unclear, ask the transaction contact to explain it using that order's details. Retain the selected item reference, applicable terms and any written response that affected your decision. Delivery timing and return arrangements also need their own current information; the drawing offers no basis for predicting either. If you change to another pad, check its account instead of assuming an earlier answer covers the replacement. After an order, keep its confirmation and associated payment record together so an inquiry can identify the specific transaction. An uncertain payment or order status needs reconciliation against those records before another payment is considered. A confirmation establishes no measured paper property and does not demonstrate that a coloring assignment is correct. You can finish every edge check in this exercise without making a purchase. Keep any later buying decision attached to the actual item and transaction evidence available at that time, including unresolved questions you still need answered.

Leave a result another reader can reproduce

Keep the vertex list, five-edge list and final assignment together. Another reader can compare each connected pair without guessing what a faint pencil mark means. If you alter a label while experimenting, date or number that proposal so its checks stay attached to the version they evaluate.

The complete result for the original graph is A1, B2, C1, D2, E3. AB, BC, CD, DE and EA all join different labels. The forced two-label attempts fail at EA under either starting choice, while this three-label assignment passes every edge. Those recorded checks establish the minimum of three for the exact graph you defined.